On straightening for Segal spaces

The straightening–unstraightening correspondence of Grothendieck–Lurie provides an equivalence between cocartesian fibrations between $(\infty, 1)$-categories and diagrams of $(\infty, 1)$-categories. We provide an alternative proof of this correspondence, as well as an extension of straightening–unstraightening to all higher categorical dimensions. This is based on an explicit combinatorial result relating two types of fibrations between double categories, which can be applied inductively to construct the straightening of a cocartesian fibration between higher categories.


Introduction
The Grothendieck construction, or unstraightening, is a standard procedure in category theory that associates to a diagram of categories F : C −→ Cat a map of categories π : Un(F ) −→ C whose fibers are precisely the values of the diagram F . This maneuver does not result in a loss of information: the unstraightening functor Un : Fun(C, Cat) Cat/C is an embedding, whose image is the subcategory of cocartesian fibrations over C and maps between them that preserve cocartesian arrows [18]. The practical consequences of this result are twofold. On the one hand, the Grothendieck construction also provides an equivalence between cocartesian fibrations over C and pseudofunctors F : C −→ Cat. From this point of view, cocartesian fibrations provide a convenient way to encode coherent diagrams of categories, which can be rectified to strict diagrams via the inverse of unstraightening, aptly called straightening. Consequently, the language of fibrations starts to play an essential role in the homotopy coherent setting of (∞, 1)-categories, where a version of straightening and unstraightening is available as well [33] (see also [43]). Indeed, many homotopy coherent diagrams of (∞, 1)-categories are most naturally described in terms of cocartesian fibrations.
On the other hand, the unstraightening of a diagram F : C −→ Cat is convenient to describe "lax constructions" with categories, stemming from the fact that Un(F ) is the oplax colimit of F [17]. For example, a lax natural transformation between two diagrams F and G can be described by a map Un(F ) −→ Un(G) over C that need not preserve cocartesian arrows (see e.g. [13,24] for a discussion in the ∞-categorical setting).
The purpose of this text is to establish a version of straightening and unstraightening for (∞, d)-categories. Recall that at the moment, the theory of (∞, d)-categories is only in the first stages of its development, in particular when compared to (∞, 1)-category theory; it is already not straightforward to compare the various different homotopy-theoretic models for (∞, d)-categories (see e.g. [46] for a textbook account and [5] for a more recent state of affairs). Nonetheless, (∞, d)-categories have been of interest in various areas of mathematics, also outside of the realm of category theory itself. For example, higher categories play a role in (derived) algebraic geometry via the six functor formalism (d = 2) [15,36] and other constructions involving correspondences [9], while in topology they arise in factorization homology [2,45] and notably in topological field theories [10,35].
Given the central place of the (un)straightening correspondence in (∞, 1)-category theory, it seems fair to say that the lack of a higher-categorical version of this correspondence has been one of the main problems in the theory of (∞, d)-categories (henceforth simply referred to as d-categories). For instance, the unstraightening procedure could be useful in the study of lax natural transformations between diagrams of d-categories and in the study of (lax) limits and colimits. Our main results establish such a procedure and show that it is essentially unique.
To this end, we define for each (d + 1)-category C two certain (non-full) subcategories Cocart d (C) ⊆ Cat d+1 /C Cart d (C) ⊆ Cat d+1 /C whose objects we refer to as d-cocartesian fibrations and d-cartesian fibrations; the two notions are equivalent under reversing the directions of morphisms in every categorical dimension. The fibers of a d-cocartesian fibration are d-categories; in particular, a 0-cocartesian fibration between 1-categories is simply a left fibration between 1-categories [33]. Furthermore, a 1-cocartesian fibration between 1-categories, viewed as 2-categories with only invertible 2-morphisms, is simply a cocartesian fibration in the usual sense. More generally, a map of (d+1)-categories π : D −→ C is roughly said to be a d-cocartesian fibration if it induces (d − 1)-cartesian fibrations on mapping d-categories and if every 1morphism in C admits enough cocartesian lifts (in an enriched sense, see Section 5 for more details). For d = 2, the notion of a 2-cocartesian fibration between (2, 2)-categories has also been considered (in a different variance) by Hermida [25] (see also [4,8]). More recently, 2-cocartesian fibrations between (∞, 2)-categories have been described in terms of inner cocartesian 2-fibrations between scaled simplicial sets [14].
Our main result is then the following: Our proof of Theorem A only relies on the straightening equivalence for left (i.e. 0cocartesian) fibrations, which we recall in Section 4.3. In particular, it provides an independent proof of the straightening-unstraightening equivalence for cocartesian fibrations of Lurie [33]. In fact, we will use an inductive argument to deduce Theorem A from a slight generalization of the 1-categorical straightening equivalence, Theorem B below.
To facilitate this inductive argument, we make use of the model for (d + 1)-categories given by iterated complete Segal spaces [5]. Recall that these are certain types of (d + 1)fold simplicial spaces C : ∆ ×d+1,op −→ S. Here the d-fold simplicial space C(0) = C(0, −) is constant and corresponds to the space of objects of C, while C(1) is itself an iterated complete Segal space, modeling the d-category of arrows in C. This Segal space model has the benefit that a diagram of d-categories indexed by a (d+1)category C can be encoded explicitly by a map of (d + 1)-fold simplicial spaces X −→ C satisfying the following two conditions [7,39]: (1) X(0) is a d-category.
We will refer to such maps as Segal copresheaves, because they should be viewed as encoding a homotopy coherent action of the (d + 1)-category C on the d-category X(0). In particular, fixing the last d simplicial coordinates yields a left fibration between 1-categories. Work of Boavida [7] gives a way to straighten each of these left fibrations, resulting in an equivalence between the 1-category of Segal copresheaves over C and the 1-category of (d + 1)-functors C −→ Cat d (see also [39] and Section 4.3). This equivalence can be thought of as a rectification procedure relating an external (Segal-style) definition of copresheaves to their internal definition as functors C −→ Cat d (cf. in particular Theorem 4.32).
Warning. Let us emphasize that one should not think of a Segal copresheaf X −→ C as some sort of cocartesian fibration: its domain is typically not a (d+1)-category (unless d = 0) and the notion of a Segal copresheaf is therefore not defined internal to the theory of (d + 1)categories. In particular, note that the fibers of a Segal copresheaf are constant in the first simplicial direction, while the fibers of a d-cocartesian fibration between (d + 1)-categories are constant in the last simplicial direction. Unfortunately, the terminology employed in [7,39] seems to suggest otherwise.
The straightening-unstraightening equivalence from Theorem A is then given by a d-step combinatorial procedure that passes from Segal copresheaves to d-cocartesian fibrations. In each step, we modify the behaviour of the map of (d + 1)-simplicial spaces X −→ C only in two consecutive simplicial directions, using the following result: Theorem B (Theorem 3.1). Let C be a double category, i.e. a bisimplicial space satisfying the complete Segal conditions in the two simplicial directions (referred to as horizontal and vertical). Then there is an equivalence of categories (2) p is a cocartesian fibration in the horizontal and a right fibration in the vertical direction. Furthermore, the fibers of D −→ C and Ψ ⊥ (D) −→ C differ simply by exchanging the horizontal and vertical directions.
The definition of the equivalence Ψ ⊥ uses rather simple combinatorics, inspired by the classical Grothendieck construction. Indeed, recall that for F : C −→ Cat, an object in the Grothendieck construction is a tuple (c, x) with c ∈ C and x ∈ F (c), and a morphism is given by the composite of a cocartesian ("horizontal") arrow f : (c, x) −→ (c ′ , f ! x), followed by a "vertical" arrow in the fiber F (c ′ ). Likewise, Ψ ⊥ (D) is a double category with the same objects as D, but with a horizontal arrow consisting of a horizontal arrow in D followed by a fiberwise vertical arrow in D.
Theorem A now follows from an inductive application of Theorem B. The lowest dimensional case, the straightening-unstraightening correspondence between 2-functors C −→ Cat 1 and 1-cocartesian fibrations over a 2-category C, is simply a special case of Theorem B: when the double category C is a 2-category, the domain and codomain of Ψ ⊥ consist precisely of Segal copresheaves and 1-cocartesian fibrations, respectively. In particular, when applied to a diagram of (1, 1)-categories F : C −→ Cat (1,1) indexed by a (1, 1)-category C, one obtains the following two-step process. We first associate to F a Segal copresheaf over C, which corresponds informally to passing from strict to pseudo-functors, after which the functor Ψ ⊥ produces the classical Grothendieck construction.
Note that Theorem A is not complete: both (d+1)-functors C −→ Cat d and d-cocartesian fibrations over C can naturally be organized into (d + 1)-categories, rather than 1-categories. In the latter case, the (d+1)-category Cocart d (C) of d-cocartesian fibrations can be realized as a subcategory of the (d + 2)-category Cat d+1 /C. Both of these (large) (d + 1)-categories furthermore depend functorially on C by restriction and base change respectively; more precisely, they can each be organized into functors of (d+2)-categories Cat op d+1 −→ CAT d+1 . We then prove the following refinement of Theorem A: Theorem C (Theorem 6.20). There is a unique natural equivalence of (d + 2)-functors Cat op d+1 −→ CAT d+1 given at a (d + 1)-category C by an equivalence of (d + 1)-categories For example, notice that the slice (d + 2)-categories Cat d+1 /C do not depend on C in a 2-functorial way in general. As one may expect, it will instead be more convenient to organize all (d + 1)-categories of d-cocartesian fibrations into a (d + 1)-cartesian fibration π : Cocart d −→ Cat d+1 . The above functor then arises as its straightening, using Theorem A (one categorical dimension higher).
Variants of Theorem A and Theorem C have appeared in the literature before. The straightening of 0-cocartesian fibrations nowadays has various proofs [7,26,33]. Theorem A for 1-cocartesian fibrations over 2-categories is the content of Lurie's scaled straightening and unstraightening theorem [13,34]. Work of Buckley [8] describes the Grothendieck construction for 2-cocartesian fibrations between (2, 2)-categories as an equivalence of 3categories. Gaitsgory and Rozenblyum prove a strengthening of Theorem C for 2-cocartesian fibrations between (∞, 2)-categories, which also allows for lax natural transformations [15,Chapter 11, Theorem 1. 1.8]; this seems to rely on some unproven statements about (∞, 2)categories appearing in Chapter 10 of loc. cit., notably about the (conjectured) model for (∞, 2)-categories in terms of lax squares. Ayala, Mazel-Gee and Rozenblyum have recently deduced this lax refinement of straightening for (∞, 2)-categories from our results and have also established a version for locally 2-cocartesian fibrations [3]. Finally, let us mention that a version of Theorem C has been hypothesized in [36], where it is used to prove a universal property for the (∞, 2)-category of correspondences in an (∞, 1)-category with fiber products.
Outline. Let us give a brief overview of the contents of this paper. Sections 2 and 3 treat Theorem B and contain our main combinatorial results: Section 2 discusses the fibrations of double categories appearing in Theorem B, and defines the functor Ψ ⊥ on objects (Section 2.3). In Section 3, we then show that Ψ ⊥ extends to an equivalence of categories (Theorem 3.1). This essentially already gives a proof of straightening and unstraightening for cocartesian fibrations of 1-categories (see Corollary 4.35).
Section 4 gives some background material on higher categories, mainly using the model of iterated complete Segal spaces. To produce examples of higher categories (e.g. of the (d + 1)-category Cat d of d-categories), we also recall the comparison to the point-set model of (strict) enriched categories (Section 4.2). Section 4.3 reviews the results of Boavida [7], relating Segal copresheaves over C to functors C −→ Cat d . For our purposes, we will need to slightly rephrase the results from loc. cit. to make them ((∞, 1)-)functorial in the base C; this requires some technical results about cartesian fibrations in the setting of categories of fibrant objects, discussed in Appendix A.
In Section 5, we give the definitions of d-cocartesian fibrations and d-cartesian fibrations. The main result here (Proposition 5.25) asserts that d-cocartesian fibrations can themselves be organized into a (d + 1)-cartesian fibration π : Cocart d −→ Cat d+1 .
Finally, Section 6 contains the proofs of our main results, Theorems A and C. We first prove a version of straightening-unstraightening at the level of 1-categories (Theorem 6.1). Using this, it follows that there is a (d + 2)-functor Cocart d : Cat op d+1 −→ CAT d+1 sending each (d + 1)-category C to the large (d + 1)-category of d-cocartesian fibrations over C. We then show that this functor is representable by Cat d (Theorem 6.20), providing the existence of the natural equivalence of Theorem C. The uniqueness of this natural equivalence follows from the results in Section 6.2.
Conventions and notation. We phrase our results as much as possible in the language of (∞, 1)-categories (with the notable exception of Sections 4.2 and 4.3, where we need to recourse to point-set models). This is mostly a matter of terminology: effectively, we only start from the (∞, 1)-category S of spaces instead of the Kan-Quillen model structure on simplicial sets to avoid having to impose fibrancy conditions at every step. In particular, we avoid any ∞-categorical machinery that (tacitly) relies on the straightening results of Lurie. Throughout the text, we will omit all prefixes "∞-". For instance, we abbreviate (∞, 1)categories simply to categories and (∞, d)-categories to d-categories, referring to ordinary categories instead as (1, 1)-categories. We write c k C for the k-core of a higher category, i.e. the k-category obtained by removing non-invertible morphisms in dimensions > k.
A subspace of a space is a (−1)-truncated map S ֒−→ T , i.e. an inclusion of path components. Likewise, by a subcategory of a d-category we will mean a (−1)-truncated map i : C ֒−→ D; inductively, this means that i induces a subspace inclusion on spaces of objects and a subcategory inclusion on mapping (d − 1)-categories. (This is in fact closer to the classical notion of a replete subcategory.) In particular, this means that specifying a subcategory of a d-category comes down to specifying a subcategory of its homotopy (d, d)-category.
Categories and d-categories will typically be denoted by C, while d-fold categories (see Section 2) will be denoted C. We have furthermore tried to differentiate these notationally from specific point-set models by enriched categories or relative categories: for example, a (d + 1)-category C can be presented by an ordinary category C strictly enriched in the (1, 1)category Cat d of fibrant objects in the d-categorical model structure (cf. Section 4.2). In particular, the (d + 1)-category Cat d of d-categories is modeled by Cat d with its canonical self-enrichment (Definition 4.22); we write Cat d = c 1 Cat d for its 1-core. Likewise, we will denote by CAT d and CAT d the (higher) categories of large d-categories.

2.1.
Recollections on double categories. Recall that a simplicial space X : ∆ op −→ S is said to be a complete Segal space if it satisfies the following two conditions [40]: (1) Segal condition: for every n ≥ 2, the natural map (2) completeness: consider the simplicial set H = ∆ [3] ∼ obtained by collapsing the 1simplex {0 ≤ 2} to a point and collapsing the 1-simplex {1 ≤ 3} to a (different) point. Then the map X(0) −→ Map sS (H, X) is an equivalence.
Recall that complete Segal spaces are a model for the homotopy theory of ((∞, 1)-)categories; we will therefore also refer to complete Segal spaces simply as (1-)categories. We will be interested in analogues of complete Segal spaces for diagrams of spaces indexed by multiple copies of the simplex category.
are complete Segal spaces. When d = 2, we will refer to C as a double category. We will denote by Cat ⊗d ⊆ Fun (∆ op ) ×d , S the full subcategory spanned by the d-fold categories.
Warning 2.3. Because we impose completeness conditions in all directions, this notion of a d-fold category is not quite the ∞-categorical analogue of that of a d-uple category in the sense of Ehresmann [12], i.e. a (strict) category object in (d − 1)-uple categories. The latter type of objects (which are sometimes also called d-fold categories) corresponds instead to the more general notion of a d-uple Segal space [22]. For instance, for a double category in the sense of Definition 2.2, the completeness conditions in the horizontal and vertical direction imply that the horizontal equivalences and vertical equivalences coincide. Many examples of double (and more generally, d-uple) Segal spaces do not satisfy this condition: as an example, one can take the double Segal space of associative algebras, with horizontal morphisms given by bimodules and vertical morphisms given by algebra homomorphisms [22]. Notation 2.4 (Opposites). Let C be a d-fold category and 1 ≤ k ≤ d. We will write C k−op for the d-fold category given by taking opposites in the k-th variable: When k = 1, we will simply write C op . If C is a double category, we will furthermore write C rev for the double category with its coordinates reflected, i.e. C rev (m, n) = C(n, m).
In the case of double categories, we will employ the following conventions: we will refer to the direction of the first copy of ∆ op as the horizontal direction and to the second copy of ∆ op as the vertical direction. Furthermore, we will use the notation −→ for horizontal arrows and and for vertical arrows.
Example 2.5. Let C and D be categories. We denote by C ⊠ D the double category given by In of d-fold simplicial spaces is monoidal for the cartesian product.
We will write Y X for the internal mapping object in d-fold simplicial spaces; the proposition implies that this is also the internal mapping object for d-fold categories.
induces a surjection on path components [1,33]. It is called a right fibration if furthermore every arrow in D is cartesian. Equivalently, it is a right fibration if the map is an equivalence, see e.g. [7,Proposition 1.7]. Taking opposite categories, one obtains the notion of a cocartesian arrow and a cocartesian (resp. left ) fibration (some alternative characterizations also appear in Section 5.1). We will be interested in the following variant of this notion for double categories: (1) The square of categories is a left fibration of categories.
(2) D(0, −) −→ C(0, −) is a cartesian fibration of categories. A strong map between (left, cart)-fibrations is a commuting square where f preserves cartesian vertical arrows. We will write Fib left,cart ⊆ Fun( [1], Cat ⊗2 ) for the subcategory of (left, cart)-fibrations and strong maps between them. Remark 2.11. The class of (left, cart)-fibrations is stable under composition and base change. Furthermore, a map D −→ * is a (left, cart)-fibration if and only if D is constant in the horizontal direction. In particular, the fibers of a (left, cart)-fibration are categories concentrated in the vertical direction (and constant in the horizontal direction).
Informally, a map of double categories p : D −→ C is a (left, cart)-fibration if the following conditions hold: (a) For every horizontal arrow α : c 0 → c 1 in C and d 0 ∈ p −1 (c 0 ), there exists a unique horizontal arrow d 0 −→ α ! (d 0 ) lifting α. Furthermore, given any square σ in C together with a lift of its left vertical arrow there exists a unique dotted liftσ as indicated (this lift includes the objects α ! (d 00 ) and α ′ ! (d 01 )).
(b) For every vertical arrow β : c 0 c 1 in D and d 1 ∈ p −1 (c 1 ), there exists a vertically p-cartesian arrow β * (d 1 ) d 1 covering α. Let us now fix a square σ in C together with an object d 01 ∈ p −1 (c 01 ). A combination of (a) and (b) then implies that σ admits a unique liftσ of the form which is vertically p-cartesian, in the sense that any other square factors uniquely overσ in the vertical direction (Remark 2.10). The resulting vertical arrow is not necessarily p-cartesian and decomposes as The first map h σ is contained in the fiber p −1 (c 10 ) and provides a natural comparison between the two ways to change fibers over the square σ, from "horizontally after vertically" to "vertically after horizontally". To illustrate Definition 2.9, let us consider the following example: Proposition 2.12. Let C and D be categories and let p = (p 1 , p 2 ) : X −→ C× D be a functor such that: (i) p 1 is a cocartesian fibration and all p 1 -cocartesian arrows in X map to equivalences in D.
(ii) p 2 is a cartesian fibration and all p 2 -cartesian arrows in X map to equivalences in C. Associated to p is a map of double categories p ′ : X −→ C ⊠ D, where X(m, n) is the space of diagrams Then p ′ is a (left, cart)-fibration of double categories.
Functors p : X −→ C × D satisfying conditions (i) and (ii) have been studied in detail in [23,24], where they are called local orthofibrations. Let us point out the close analogy between the discussion preceding Proposition 2.12 where the first map is the inclusion of the wide subcategory spanned by the q-cocartesian arrows. In particular, this shows that each X(−, n) −→ C ⊠ D (−, n) is a left fibration, as desired.
Remark 2.14. This remark only serves to motivate Definition 2.9, and will not be used in the rest of the text (making it more precise would require a form of straightening). Elaborating on the discussion following Definition 2.9, one can informally think of a (left, cart)-fibration p : D −→ C as a certain type of diagram of categories indexed by C. Indeed, by Remark 2.11, the fibers of p are all categories (in the vertical direction). Furthermore, each horizontal arrow α : c 0 −→ c 1 induces a map of categories α ! : D c0 −→ D c1 and each vertical arrow β : c 0 c 1 induces a map of categories β * : D c1 −→ D c0 . Finally, each diagram σ : [1, 1] −→ C gives rise to a square of categories and functors commuting up to a natural transformation In other words, a (left, cart)-fibration p : D −→ C should arise as the unstraightening of a diagram of double categories C 2−op −→ Cat oplax , where Cat oplax is a certain double category whose objects are categories, horizontal and vertical morphisms are functors and squares are oplax squares  For example, one can also associate a (cocart, right)-fibration to a two-variable fibration (p 1 , p 2 ) : X −→ C × D as in Proposition 2.12, by considering maps σ as in (2.13) that send each {i} × [n] to p 2 -cartesian arrows in X. Remark 2.17. Similarly to Remark 2.14, one can informally think of a (cocart, right)fibration p : D −→ C as a certain diagram of categories indexed by C. Indeed, in this case all fibers of p are categories in the horizontal direction (and constant in the vertical direction) and again each horizontal arrow α : c 0 −→ c 1 induces a map of categories α ! : D c0 −→ D c1 and each vertical arrow β : c 0 c 1 induces a functor β * : D c1 −→ D c0 . For the behaviour on a square σ : [1, 1] −→ C, we can repeat the analysis following Definition 2.9: for each such square and a lift d 10 ∈ p −1 (c 10 ), there exists a lift to a square in D which is p-cocartesian in the horizontal direction. The left vertical arrow is p-cartesian but the right vertical arrow is not in general, so that one again obtains a natural map In other words, one can think of a (cocart, right)-fibration p : D −→ C as the unstraightening of a map of double categories C 2−op −→ Cat oplax , where Cat oplax is the double category described informally in Remark 2.14.
2.3. The reflection of a (left, cart)-fibration. We will now turn to the main combinatorial construction of this text: we will associate to each (left, cart)-fibration p : D −→ C a (cocart, right)-fibration Ψ ⊥ (D) −→ C, which we will refer to as the reflection of p. In Section 3, we will study the functoriality of this construction in more detail.
Before we begin, note that at a heuristic level, Remark 2.14 and Remark 2.17 show that (left, cart)-fibrations and (cocart, right)-fibrations over a double category C encode the same kind of data: both should be considered as unstraightened versions of the datum of a map of double categories Cat oplax . One should therefore be able to straighten a (left, cart)-fibration to such a map of double categories and then unstraighten it to obtain a (cocart, right)-fibration. Instead of making this more precise, we will give a purely combinatorial description of (what should be) the composite functor Ψ ⊥ , inspired by the explicit formula for the Grothendieck construction and the combinatorics appearing in [23]. Informally, the functor Ψ ⊥ will first remove the vertical arrows in each fiber of p : D −→ C, so that only the cartesian vertical arrows remain, and then replaces them by fiberwise horizontal arrows instead (which can be composed with the cocartesian horizontal arrows already present in D).
Let us start by considering the following construction of a double category out of a (left, cart)-fibration, which will appear repeatedly in the text: Lemma 2.18. Let p : D −→ C be a map of double categories such that each D(−, n) −→ C(−, n) is a left fibration. Consider the bisimplicial space X whose space of (m, n)-simplices is given by the space of commuting squares π hor ×id p Then X is a double category.
Remark 2.20. Informally, X is a double category with the same objects as D, whose squares are given by diagrams in D of the form where the right square is a commuting square of vertical arrows in which the top and bottom arrow are sent to degenerate arrows in C. In other words, we keep the same vertical arrows but replace the horizontal arrows by formal composites β • α : · −→ · · of a horizontal and a fiberwise vertical arrow. Diagrams of the above form have an evident vertical composition. For the composition of horizontal arrows, consider the concatenation of two formal composites · · · · · α1 β1 α2 β2 Then β 1 and α 2 fit into an essentially unique (1, 1)-cell of D that covers the vertically degenerate (1, 1)-cell p(α 2 ) in C. The other two sides of this square give a formal composite β ′ 1 • α ′ 2 : · −→ · ·. The composed horizontal arrow in X is then given by the formal Proof. To see that X is a category in the vertical direction, we use that Cat ⊗2 is cartesian closed (Proposition 2.8): the simplicial space X(m, −) can then be identified with the To see that each X(−, n) is a category, it suffices to treat the case n = 0: indeed, the general case then follows by considering the functor D [0,n] −→ C [0,n] , which is still a left fibration in the horizontal direction. Unraveling the definitions shows that the Segal condition for X(−, 0) translates into the unique lifting problem In other words, given a solid diagram in D of the form x 00 x 01 x 02 . . .
x 1m . . . . . . x 00 x 01 x 02 x 03 x 11 x 12 x 13 x 22 x 23 x 33 in which x 00 −→ x 02 and x 02 x 22 are degenerate and x 11 −→ x 13 and x 13 x 33 are degenerate, and whose image in C consists only of degenerate arrows (since C is complete). We need to show that the above diagram is degenerate both in the horizontal and the vertical direction. Using that each D(−, n) −→ C(−, n) is a left fibration, one sees that the above diagram is degenerate in the horizontal direction. Furthermore, the rightmost column is degenerate in the vertical direction, since both x 03 x 23 and x 13 x 33 are degenerate and D(0, −) is a category. Definition 2.21. Let p : D −→ C be a (left, cart)-fibration and let X be the double category from Lemma 2.18. We will denote by Ψ ⊥ (D) ⊆ X the bisimplicial subspace whose (m, n)-simplices correspond to diagrams (2.19)   Proof. We verify each of the conditions, starting with the fact that Ψ ⊥ (D) is a double category.
Segal conditions. Recall the double category X from Lemma 2.18. Since each Ψ ⊥ (D)(−, n) ⊆ X(−, n) is a full subcategory of the category X(−, n), it follows that Ψ ⊥ (D)(−, n) is a category (i.e. complete Segal space) itself. In the other direction, note that there is a pullback square of simplicial spaces x 00,0 x 01,0 x 00,1 x 01,1 D x 11,0 there exist unique objects x 00,0 , x 01,0 , x 11,0 together with a dashed extension as indicated, such that the two arrows x 00,0 x 00,1 and x 11,0 x 11,1 are cartesian. To see this, first take these two arrows to be the (unique) cartesian lifts of their image in C. Then the top square can be filled uniquely using that D −→ C is a left fibration in the horizontal direction. The right square (living entirely in the vertical direction) can then be filled uniquely since x 11,0 x 11,1 was cartesian.
Horizontal cocartesian fibration. Note that the space Ψ ⊥ (D)(1, 0) of horizontal arrows in Ψ ⊥ (D) is the space of diagrams in D of the form x y z covering a horizontal arrow in C. Let us say that a horizontal arrow in Ψ ⊥ (D) is marked if the corresponding vertical map y z is an equivalence in D. We claim that all marked horizontal arrows in Ψ ⊥ (D) are Ψ ⊥ (p)-cocartesian. In other words, we have to find a unique diagonal lift for every diagram of categories where the top map sends 0 −→ 1 to a marked arrow. Unraveling the definitions, this corresponds to a solid diagram in D covering a horizontal 2-simplex in C x 00 x 01 x 02 x 11 x 12 where the map x 01 x 11 is an equivalence, of which one has to find a unique dashed extension as indicated. To find the desired unique extension, using that D −→ C is a left fibration in the horizontal direction, it follows that there is a unique way to extend the square making the top triangle (living purely in the horizontal direction) commute. Since x 01 x 11 was an equivalence, the resulting vertical arrow x 02 x 12 is an equivalence as well. It then follows that there is a unique vertical map x 12 x 22 with a filling of the right triangle (purely in the vertical direction). We conclude that all marked arrows in Ψ ⊥ (D) are cocartesian. Finally, for every horizontal arrow f : c −→ c ′ in C and a lift x ∈ Ψ ⊥ (D) c , there exist a marked liftf : x −→ x ′ , using that D(−, 0) −→ C(−, 0) was a left fibration: indeed, one can takef to correspond to the diagram

Reflecting fibrations of double categories
This section provides a more detailed analysis of the reflection of (left, cart)-fibrations between double categories (Definition 2.21). More precisely, our goal will be to prove our main technical result: At the level of objects, the functor Ψ ⊥ is given by Definition 2.21. The functor Ψ ⊤ is defined analogously, taking opposites and reversing the roles of the horizontal and vertical directions. To extend Ψ ⊥ and Ψ ⊤ to functors, it will be convenient to describe the ∞-categories of (left, cart)-fibrations and (cocart, right)-fibrations in terms of marked bisimplicial spaces (Section 3.1). This allows one to describe the functor Ψ ⊥ very explicitly in terms of combinatorial data (Section 3.2). Finally, we will provide explicit natural equivalences exhibiting Ψ ⊥ and Ψ ⊤ as mutual inverses (Section 3.3). We will do this by explicitly writing down a zigzag of equivalences connecting their composition to the identity.
Remark 3.2. In model categorical terms, both functors Ψ ⊥ and Ψ ⊤ arise as right Quillen functors between the arrow categories of marked bisimplicial spaces. In particular, all of our arguments can be reformulated without difficulties into such model-categorical terms.
3.1. Marked bisimplicial spaces. For technical reasons, it will be convenient to give a description of the category Fib left,cart using markings, analogous to the model for cartesian fibrations using marked simplicial sets in [33].
↓ denote the category obtained from ∆ ×2 by freely adding a factorization of the vertical degeneracy map [0, 1] −→ [0, 0] as follows: A presheaf X : (∆ ×2 ↓ ) op −→ S is said to be a vertically marked bisimplicial space if the map X([0, 1] ♯ ) −→ X(0, 1) is a subspace inclusion. We will write bisS ↓ ⊆ Fun(∆ ×2,op ↓ , S) for the full subcategory spanned by the vertically marked bisimplicial spaces, and typically use the same symbol for a vertically marked bisimplicial space and its underlying bisimplicial space. Remark 3.4. The forgetful functor bisS ↓ −→ bisS induces a subspace inclusion Map bisS (X, Y ) whose image consists of the path components of maps of bisimplicial spaces f : X −→ Y with the property that X(0, 1) −→ Y (0, 1) preserves the subspace of marked vertical arrows.
Definition 3.5. A vertically marked double category is a vertically marked bisimplicial space whose underlying bisimplicial space is a double category. We will write Cat ⊗2 ↓ ⊆ bisS ↓ for the full subcategory of vertically marked double categories.
Likewise, we write Cat ⊗2 → ⊆ bisS → for the full subcategory of horizontally marked bisimplicial spaces spanned by the horizontally marked double categories. Notation 3.6. As usual, the forgetful functor Cat ⊗2 ↓ −→ Cat ⊗2 admits a fully faithful right adjoint, sending a double category C to the vertically marked double category C ♯ where all vertical arrows are marked. We will write also write C ♯ for C with all horizontal arrows marked; it will always be clear from the context whether we mark horizontal or vertical arrows. (1) Let Fun ♯ [1], bisS ↓ be spanned by the maps Y −→ X where X is maximally marked, i.e. every [0, 1]-simplex of X is marked.
(3) Let Fib left,cart be spanned by the maps p : D ♮ −→ C ♯ between marked double categories, whose underlying map of double categories is a (left, cart)-fibration and where a vertical arrow in D ♮ is marked if and only if it is p-cartesian. The codomain projections (forgetting the maximal marking) then define a diagram of cartesian fibrations and maps preserving cartesian arrows ) which is an equivalence onto the subcategory whose objects are (left, cart)-fibrations and whose morphisms are strong morphisms (Definition 2.9).
Proof. By Remark 3.4, the forgetful functor Fun [1], Cat ⊗2 ↓ ) −→ Fun [1], Cat ⊗2 ) induces inclusions of path components on mapping spaces, whose images consist of those commuting squares of double categories such that all maps preserve marked arrows. The result follows immediately from this. Remark 3.9. The full subcategory Fib left,cart ֒−→ Fun [1], bisS ↓ can be obtained as a left Bousfield localization (and in particular is a presentable category). One can also present Fib left,cart in terms of model (1, 1)-categories, as a left Bousfield localization of the injective model structure on functors ∆ ×2,op ↓ −→ sSet (where sSet carries the Kan-Quillen model structure). All arguments appearing below can be carried out in this setting as well.
, an additional set of path components of marked vertical edges. We will write S[1, 0] + for the resulting vertically marked double category. Associated to this data is a natural map of cartesian fibrations

ΨS codom codom
To construct this diagram, observe that the codomain projection admits a left adjoint L : bisS −→ Fun ♯ ([1], bisS ↓ ), sending C to ∅ −→ C ♯ ; the same holds in the horizontally marked case. Now consider the commuting squares In the left square, h is the Yoneda embedding and σ is the functor sending This induces the commuting square of left adjoint functors on the right, using that h and σ induce unique colimit-preserving functors out of the corresponding presheaf categories [11], [33, Theorem 5.1.5.6]. Let us write Ψ S for the right adjoint to Φ S . Explicitly, let p : Y −→ X ♯ be a map of vertically marked bisimplicial spaces whose target is maximally marked. Then Ψ S (p) : Ψ S (Y ) −→ X ♯ can be described as follows: the underlying bisimplicial space of Ψ S (Y ) has (m, n)simplices determined by Furthermore the space of marked [1, 0]-simplices in Ψ S (Y ) is the subspace consisting of those maps S[1, 0] −→ Y that also preserve the additional marked arrows from (b). In particular, this implies that Ψ S sends vertically marked bisimplicial spaces into horizontally marked bisimplicial spaces (i.e. marked edges form a subspace). We therefore obtain the desired commuting diagram (3.11). Furthermore, it follows immediately from Equation (3.12) that Ψ K preserves cartesian morphisms: for every Y −→ X ♯ and X ′ −→ X, we have an equivalence Construction of the functors. We will use Construction 3.10 to define the functor Ψ ⊥ , as follows:  [1], where in addition the vertical arrow 01 11 is marked.
Using this, Construction 3.10 produces a functor over bisS Proposition 3.14. Let p : D ♮ −→ C ♯ be a (left, cart)-fibration with its natural marking by the p-cartesian vertical arrows. Then Ψ K (p) : Ψ K (D) −→ C ♯ coincides with the (cocart, right)-fibration Ψ ⊥ (p) from Definition 2.21, with its natural horizontal marking. Consequently, Ψ K restricts to a map of cartesian fibrations that we will denote by Proof. Unraveling the definition, one sees that the map of bisimplicial spaces Ψ K (D) −→ C is precisely given by the reflection Ψ ⊥ (p) : Ψ ⊥ (D) −→ C from Definition 2.21; in particular, it is a (cocart, right)-fibration by Proposition 2.22. It remains to identify the marked horizontal arrows in Ψ K (D ♮ ); these correspond to diagrams x −→ y z in D with the property that y z is a marked arrow covering an equivalence in C. This just means that y z is an equivalence, so that the marked horizontal arrows coincide with the marked horizontal arrows introduced in the proof of Proposition 2.22. We have already seen there that these marked arrows are precisely the cocartesian horizontal arrows.
In particular, restricting to the fibers over a double category C, we obtain a functor Ψ ⊥ : Fib left,cart (C) −→ Fib cocart,right (C) that we will refer to as the reflection functor. To motivate the terminology, let us investigate more precisely the behaviour of Ψ ⊥ when C is a space: In particular, at the level of fibers the reflection functor Ψ ⊥ simply exchanges the horizontal and vertical direction.
where the second map is the vertical projection (see Example 2.6). This map sends the marked vertical arrows in K[m, n] from Construction 3.13 to degenerate edges. For each D ∈ Fib left,cart , we therefore obtain a natural transformation of double categories, given in degree (m, n) by .
. To see that this is an equivalence, note that the domain and codomain are both constant in the vertical direction and complete Segal spaces in the horizontal direction. In degree (0, 0), the above is an equivalence since K[0, 0] −→ [0, 0] is an isomorphism. In degree (1, 0), it suffices to note that the map π ver : Ar[1] −→ [0, 1] induces an equivalence on mapping spaces into a double category of the form [0] ⊠ C.
One can obtain variants of the functor Ψ ⊥ for different kinds of fibrations, by taking opposites in the horizontal or vertical direction and exchanging horizontal and vertical directions. (1) There is a functor Ψ ⊤ : Fib cocart,right −→ Fib left,cart , obtained by conjugating Ψ ⊥ with the functor sending C → (C rev ) (1,2)−op . Unraveling the definitions, one sees that Ψ ⊤ = Ψ K ′ is defined as in Construction 3.10 (with the roles of horizontal and vertical reversed), but this time using the maps Here we mark the horizontal arrows in where we mark all horizontal arrows.
(2) There is another functor Ψ † : Fib cocart,left −→ Fib left,cocart , obtained by conjugating Ψ ⊥ with the functor sending C → (C rev ) 1−op . Unraveling the definitions, one sees that We will show that these are mutually inverse, up to homotopy. More precisely, we will prove that for every (left, cart)-fibration D −→ C, there is a natural zig-zag of equivalences between (left, cart)-fibrations over C To motivate the precise combinatorial constructions that will follow, let us already disclose that a typical (1, 1)-cell in each of these four double categories will correspond to a diagram in D of the following form: Here • denotes a p-cartesian vertical arrow in D and denotes a map that is sent to a degenerate arrow in C. The rightmost diagram can be obtained by explicitly unraveling the definition of the composite functor Ψ ⊤ • Ψ ⊥ . The maps ζ * and θ * will then be the evident equivalences which simply add degenerate squares on the right, resp. on the top.
The map η * takes vertical compositions and has an inverse given informally as follows (cf. the proof of Proposition 2.22). First, one decomposes the left and right vertical maps into fiberwise maps followed by p-cartesian maps. The middle column is then uniquely determined by the left column and the right half of the diagram is determined uniquely because the right vertical map was p-cartesian. Notice that the map f need not be pcartesian; this is why we need to add a degenerate square using ζ * before we can decompose the left and right vertical arrow.
Let us now start by defining the endofunctors Ψ A , Ψ B : Fib left,cart −→ Fib left,cart more precisely, using (the vertically marked version of) Construction 3.10.
, we mark the following vertical arrows: Then the natural transformations ζ * and η * above are equivalences of marked bisimplicial spaces. In particular, By Lemma 2.18, X is double category and one sees that each simplicial subspace is the inclusion of a full subcategory. In the horizontal direction, the proof of Lemma 2.18 shows that the simplicial subspace Ψ A (D ♮ )(−, n) ֒−→ X(−, n) satisfies the Segal conditions and is an equivalence on spaces of objects. This implies that Let us now show that the map of marked double categories ζ * : is an equivalence on marked vertical arrows. It then remains to verify that the map on (1, 1)-simplices is an equivalence. Unraveling the definitions, a (1, where the right square covers a vertical arrow in C. The map ζ * : is simply the inclusion of the diagrams of the above form where the right square is degenerate; this is an equivalence by completeness of D. where P denotes the pullback. Since the bottom map is the inclusion of a wide subcategory (as we have seen above), P is a wide subcategory of X x 00 x 01 . . . x 0n Here each of the slices labeled by x, y and z is a diagram in D(0, −) parametrized by z ii on the diagonal then extends uniquely to a natural transformation y ij z ij , since the target is the relative right Kan extension of its restriction to the diagonal. We conclude that η * : is an equivalence on the underlying bisimplicial spaces. Finally, it identifies the marking since Our next goal is to describe the composite functor Ψ ⊤ • Ψ ⊥ in terms of Construction 3.10.

Construction 3.21.
For each m, n ≥ 0, let I[m, n] denote the (1, 1)-category whose objects are quadruples of maps of linear orders Given two such quadruples, a map between them is given by a quadruple of maps We view T [m, n] as a marked bisimplicial space by marking those (0, 1)-simplices corresponding to a quadruple (α, β, γ, δ) (with s = 0, t = 1) such that either: (a) the map γ : where all vertical arrows are marked.
Proof. Recall from Construction 3.10 that Ψ K ′ arises as the right adjoint to a functor Unraveling the definitions, the marked (1, 0)-simplices of K ′ [m, n] correspond precisely to tuples of α : We can apply the same reasoning to the left adjoint Φ K • Φ K ′ : this functor preserves colimits and sends In other words, it is precisely given by T [m, n]; unraveling the definitions shows that the marking coincides with that of Construction 3.21. Likewise, Proof (of Theorem 3.1). By Proposition 3.14 (and Variant 3.16), we have functors To see that they are mutually inverse, it suffices to prove that Ψ ⊤ • Ψ ⊥ ≃ id. Indeed, in this case, conjugating by the functor We construct the desired natural equivalence as a zig-zag Here the first two equivalences follow from Proposition 3.19 .22), consider the map Example 2.6). Combining these two, we obtain a natural map For any (left, cart)-fibration p : D ♮ −→ C ♯ , restriction along θ therefore defines a natural map of marked bisimplicial spaces

Unraveling the definitions, a vertical arrow in Ψ
Here first two maps live in the fiber over c and the vertical arrow d ′′

Recollections on higher categories
In the remainder of the text, we will use Theorem 3.1 to establish a version of straightening and unstraightening for cocartesian fibrations of d-categories. To facilitate this, we will start in this section by recalling some preliminary results on d-categories; our default model for d-categories will be in terms of iterated Segal spaces [5]. Section 4.2 provides a reminder on point-set models for d-categories; its main purpose is to allow us to construct explicit examples of higher categories (such as the (d + 1)-category of d-categories) from enriched categories. Sections 4.3 and 4.4 describe the theory of copresheaves (of d-categories) on (d + 1)categories; this is mostly a recollection of the work of Boavida [7]. For our purposes, it will be convenient to reformulate the results of loc. cit. as Proposition 4.40: this provides a natural equivalence between the external definition of copresheaves over a (d + 1)-category C, in terms of certain fibrations of (d + 1)-fold simplicial spaces X −→ C, and the internal definition of copresheaves, as maps of (d + 1)-categories C −→ Cat d .

4.1.
Higher categories via Segal spaces. Recall the iterated Segal space model for dcategories [5]: constant. We will write Cat d ֒−→ Cat ⊗d for the full subcategory spanned by the d-categories.
Remark 4.2. The above definition of a d-category is a priori more restrictive than the usual one from [5]: we ask all C( n k−1 , −, n d−k ) to be complete, while this is usually only required if n d−k = 0 d−k . It follows from [30, Lemma 2.8] that the two definitions coincide, essentially by an Eckmann-Hilton type argument.
If C is a d-category, recall that the (constant (d − 1)-fold simplicial) space C(0) describes the space of objects of C. Given two objects x 0 , x 1 ∈ C, we will write for the mapping (d − 1)-category between them. For any triple of objects x 0 , x 1 , x 2 , there is a composition map, determined by the commuting square Here the left vertical map is an equivalence by the Segal conditions.
To efficiently deal with the additional constancy conditions appearing in Definition 4.1, let us introduce the following notation: that exhibit the category of presheaves on ∆ I both as a localization and a colocalization of the category of presheaves on ∆ × I. (1) ∆ I is a (1, 1)-category with a terminal object.
(2) For any X : ∆ × I op −→ S, consider the natural map of presheaves on ∆ × I where i * denote the right Kan extension along i : {0} × I ֒→ ∆ × I and i * X −→ X(0, t) denotes the natural map from the presheaf X(0, −) on I to the constant presheaf with value X(0, t). This map exhibits the counit of the adjoint pair (λ * , λ * ).
In fact, in this case ∆ I can be identified with a (non-full) subcategory of ∆ ≀ I, given by the image of the functor δ : ∆ × I −→ ∆ ≀ I from [6].
For assertion (1), consider the following (homotopy) pushout of (space-valued) presheaves Here the top map is given pointwise by the inclusion of the subset of maps Since i is fully faithful, i * i * ≃ id and the restriction of Y along i : {0} × I −→ ∆ × I coincides with the constant presheaf with value X(0, t). This means that Y is contained in the essential image of λ * , or equivalently, that Y is a λ * λ * -colocal object.
It therefore remains to verify that the projection Y −→ X is a colocal equivalence, i.e. that it is sent to an equivalence by λ * . Since Y −→ X is the base change of the map i * (X(0, t)) −→ i * i * X, it suffices to to verify that λ * i * (X(0, t)) −→ λ * i * i * X is an equivalence. By part (1), the functor λi decomposes as I −→ * −→ ∆ I , where the second functor is the inclusion of the terminal object. The result then follows from the fact that the natural map is an equivalence since t was the terminal object of I.
By construction, restriction along δ d determines a fully faithful inclusion We will systematically omit δ * d and simply identify presheaves on ∆ d with d-fold simplicial spaces using this inclusion. Likewise, we can identify presheaves on ∆ d with simplicial object in presheaves on ∆ d−1 which are constant in simplicial degree 0, etcetera.
In these terms, we have a pullback square of fully faithful functors Since the bottom and the right functors are right adjoint functors between presentable categories, the inclusion Cat d −→ Cat ⊗d admits a left adjoint. It also admits a right adjoint, by the following observation: Then r d maps the full subcategory Cat ⊗d to Cat d . Proof. We proceed by induction, the case d = 1 being clear. Let us now take d > 1 and assume that r d−1 sends (d−1)-fold categories to d-categories. Suppose that X : ∆ ×d,op −→ S is a d-fold category and factor the map δ d as The right Kan extension along the functor id × δ d−1 sends X to the simplicial object is then a (d − 1)category by inductive hypothesis and the simplicial object X ′ satisfies the complete Segal conditions since r d−1 is a right adjoint.
Next, we compute r d (X) = λ * (X ′ ) using Lemma 4.4 (2). Denoting by i : at least after precomposing with the localization λ : Here the map . This implies that r d (X) satisfies the Segal condition in the first simplicial direction (as the spine inclusions induce bijections on objects). Furthermore, each r d (X)(m) is a (d − 1)-category, since all terms in the above pullback diagram are (d − 1)-categories. It remains to verify the completeness condition. By Remark 4.2, it suffices to verify the completeness condition after setting the last variables equal to 0 d−1 . But in that case the above formula shows that r d (X)(−, 0 d−1 ) ≃ X ′ (−, 0 d−1 ), for which we already verified that it was a complete Segal space. Remark 4.8. In fact, one sees that for any d-fold category C, the counit map r d (C) ֒−→ C is a d-fold simplicial subspace, given in degree (m 1 , . . . , m d ) by the path components x satisfying (inductively) the following two conditions: it is a d-category with two objects 0, 1 with trivial endomorphisms and Map [1] A (0, 1) = A.
If C is a d-category, then the d-category can be described as follows. Its space of objects is given by the space of maps α : [1] A −→ C, i.e. of tuples α 0 , α 1 ∈ C and α : A −→ Map C (α 0 , α 1 ). Unraveling the definitions, one sees that the (d − 1)-category of maps between two such α, β : [1] A −→ C is given by the pullback  , 0), which we will refer to as the (d − 1)-core.

Point-set models.
To produce examples of d-categories, it will be more useful to think of d-categories as categories enriched in (d − 1)-categories. This is probably best done using the theory of enriched ∞-categories [16,28]. We will take a more rigid, model-categorical approach instead (which has the advantage of not relying on Lurie's (un)straightening equivalence). Recall from Notation 4.5 how we view presheaves on ∆ d as d-fold simplicial spaces using the localizations (1) Each X(n) : ∆ d−1 −→ S is a (d − 1)-category (and X(0) is a space).
Remark 4.13. The category of d-categories can be identified with the full subcategory of the category of d-categorical algebras, on those d-categorical algebras that are furthermore complete [16,34]. The inclusion of d-categories into d-categorical algebras admits a left adjoint L DK , which localizes at the class of Dwyer-Kan equivalences, i.e. maps Y −→ X with the following two properties (see loc. cit.): (1) fully faithfulness: is an equivalence of (d − 1)-categories.
In particular, for each d-categorical algebra X, the map X −→ L DK (X) is a Dwyer-Kan equivalence. In fact, the explicit construction shows that X −→ L DK (X) is not just essentially surjective, but that X(0) −→ L DK (X)(0) is already surjective on π 0 [34, Proposition 1.2.27].
Recall that there is a localization functor sSet −→ S inverting the weak equivalences for the Kan-Quillen model structure. We will say that a functor X : (1) The cofibrations are the monomorphisms, and in particular every object is cofibrant.
(2) An object X is fibrant if and only if it is fibrant in the injective model structure and it is a d-category model. (3) A weak equivalence between d-category models is a levelwise weak equivalence of simplicial sets (in the Kan-Quillen model structure). We will write Cat d for the full subcategory of sPSh(∆ d ) on the injectively fibrant d-category models.   • X satisfies the Segal conditions in the first simplicial variable.
We will say that a map Y −→ X is a Dwyer-Kan equivalence if the corresponding map of categorical algebras is a Dwyer-Kan equivalence as in Remark 4.13. Furthermore, we will say that a map Y −→ X is an isofibration if: (1) Each Y (n) −→ X(n) is a fibration in Cat d−1 , i.e. an injective fibration of simplicial presheaves on ∆ d−1 .
(2) The induced map on homotopy categories ho(Y ) −→ ho(X) is an isofibration. With these classes of maps, CatAlg d becomes a category of fibrant objects as in Appendix To factor a map Y −→ X in CatAlg d into a weak equivalence followed by a fibration, one can simply factor it into a weak equivalence followed by a fibration with respect to the d-categorical model structure on sPSh(∆ d ); such fibrations are isofibrations by the right lifting property against {0} −→ H, where H is the simplicial set from Section 2. Map C (c n−1 , c n ) × · · · × Map C (c 0 , c 1 ).
In particular, N(C)(0) is simply the set of objects of C. We will say that a map of enriched categories is a Dwyer-Kan equivalence, resp. an isofibration, if its image under the nerve functor is such. This makes Cat(

4.3.
Copresheaves. If C is a (d+1)-category, we define the (d+1)-category of copresheaves (of d-categories) of C to be the functor (d + 1)-category Fun d+1 (C, Cat d ), i.e. as the internal mapping object in the category of (large) (d + 1)-categories. This (d + 1)-category can be presented explicitly in terms of the enriched categories model for (d + 1)-categories: In particular, using the point-set model of Cat d -enriched categories, a copresheaf C −→ Cat d can be modeled by an enriched functor C −→ Cat d . In addition to this 'internal' definition of copresheaves, one can also describe copresheaves externally in terms of Segal objects, following [7] (see also [39] for an in-depth discussion): Definition 4.25. Let C be a (d + 1)-categorical algebra. A map of (d + 1)-fold simplicial spaces X −→ C is said to be a Segal copresheaf if the following two conditions hold: (2) For each [n] ∈ ∆, there is a pullback square of d-categories Let us write coPSh Seg d ⊆ Fun [1] × ∆ ×d+1,op , S for the full subcategory spanned by the Segal copresheaves X −→ C where C is a (d + 1)-category. The codomain projection π : coPSh Seg d −→ Cat d+1 is a cartesian fibration, since the pullback of a Segal copresheaf X −→ C along a map of (d + 1)-categories C ′ −→ C is again a Segal copresheaf. We will write coPSh Seg d (C) for the fiber over C. Warning 4.26. When d = 0, the domain of a Segal copresheaf over a 1-category is itself a 1-category [7,Corollary 1.19]; in this case, Segal copresheaves can be identified with left fibrations (or, in the terminology of Section 5, 0-cocartesian fibrations). Note that for d ≥ 1, the domain of a Segal copresheaf X −→ C over a (d + 1)-category is typically not itself a (d + 1)-category: if the domain were a (d + 1)-category, then all the fibers of the fibration would be spaces.
In the remainder of this section, we will recall the work of Boavida [7], which shows that the category of Segal copresheaves coPSh Seg d (C) coincides with the 1-category underlying Fun d+1 (C, Cat d ). We will make use of point-set models to compare to Proposition 4.24:  Construction 4.29. Let X : ∆ op −→ Cat d be an object of CatAlg d+1 , i.e. X satisfies the Segal conditions and X(0) ∈ sPSh(∆ d ) is constant on a simplicial set. The category sPSh(∆ × ∆ d ) /X can be endowed with the projective covariant model structure, in which: The identity functor determines a Quillen equivalence to the injective covariant model structure, defined similarly but as a left Bousfield localization of the injective model structure on sPSh(∆ × ∆ d ) /X . We will write coPSh Seg,proj  Proof. We verify conditions (a) and (b) of Lemma A.2: for each Dwyer-Kan equivalence f : X −→ X ′ , there is an adjoint pair f ! : sPSh(∆ × ∆ d ) /X ⇆ sPSh(∆ × ∆ d ) /X ′ : f * . This is both a Quillen pair for the projective covariant and injective covariant model structure; in particular f ! preserves all weak equivalences. It now follows from [7, Proposition 5.5] that (f ! , f * ) is a Quillen equivalence: indeed, the result from loc. cit. shows that the derived unit and counit are already equivalences pointwise in ∆ d . This immediately gives condition (a), while (b) follows from Remark A.3.      C op × C.
The Segal conditions on C imply that this is a Segal copresheaf. By Corollary 4.33, this corresponds to a (d + 1)-functor that we will denote by This functor is easily described using point set models: if C is an enriched category, then the above map of simplicial spaces is modeled by the object Tw(N(C)) −→ N(C) op × N(C) in coPSh Seg,proj (N(C)). Unraveling the definitions, this left fibration is exactly the image under N : Fun enr (C op × C, Cat d ) −→ coPSh Seg,proj (N(C)) of the (strict) mapping space functor We will use the language of Example 4.37 to partially address the functoriality of Corollary 4.33: Notation 4.39. For a regular uncountable cardinal κ, let us write coPSh Seg d (κ) for the category of Segal copresheaves X −→ C whose fibers are essentially κ-small. The codomain projection coPSh Seg d (κ) −→ Cat d+1 is a cartesian fibration. Taking the subcategory with only cartesian morphisms, we obtain a right fibration π : coPSh Seg d (κ) cart −→ Cat d+1 . Note that there is a canonical element u ∈ coPSh Seg d (κ) cart , corresponding under the equivalence of Corollary 4.33 to the identity functor Cat d (κ) −→ Cat d (κ) on the small (d + 1)-category of κ-small d-categories.
Proposition 4.40. The element u : * −→ coPSh Seg d (κ) defines a representation of the right fibration coPSh Seg d (κ) cart −→ Cat d+1 . In other words, for every (d + 1)-category C, there is a 1-functorial equivalence between the space of (d + 1)-functors C −→ Cat d (κ) and the space of Segal copresheaves X −→ C with κ-small fibers. Remark 4.41. For later purposes, we will mostly be interested in this result in the large setting, with κ being the supremum of all small cardinals: in that case it provides a 1-functorial equivalence between the spaces of (d + 1)-functors C −→ Cat d and Segal copresheaves X −→ C with small fibers over a large (d + 1)-category C.
The proof uses the following well-known criterion for being a representation (cf. [  Proof. Assuming x defines a representation of X −→ C, we can model C by an enriched category and identify x with the canonical map id c : * −→ N (Map C (c, −)). One easily verifies that id c defines an initial object in each degree n d .
Conversely, suppose that x defines an initial object in each X(−, n d ). Note that there is a fully faithful inclusion coPSh Seg d ֒−→ Fun(∆ d , coPSh Seg 0 ) which admits a left adjoint and preserves cartesian arrows. This implies that (4.38) defines a cocartesian arrow in coPSh Seg d as soon as it does in Fun(∆ d , coPSh Seg 0 ). It therefore suffices to verify that each * −→ X(−, n d ) defines a cocartesian arrow in the category of Segal copresheaves over 1categories. This reduces everything to the case d = 0, where we can consider the diagram * X X * X C.
x id id p x p One easily verifies that the right square defines a cocartesian arrow in the category coPSh Seg 0 of left fibrations. On the other hand, since x ∈ X is an initial object, the left square represents the terminal copresheaf on X, i.e. it also defines a cocartesian arrow in coPSh Seg 0 . The composite is then a cocartesian arrow as well, as desired. Here ψ * is fully faithful with essential image consisting of presheaves sending W to equivalences. We have to show that ψ ! (u) ≃ * (where we omit the Yoneda embedding from the notation). To see this, consider for each small cardinal λ ≥ κ the full Cat d -enriched subcategory U λ,n ⊆ Fun([n], Cat d ) spanned by sequences of trivial fibrations A 0 ։ · · · ։ A n where each A i is λ-small and homotopy equivalent to a κ-small object. Each U λ,n comes with a canonical enriched functor p 0 : U λ,n −→ Cat d sending (A 0 −→ . . . −→ A n ) to A 0 . Let us write u λ,n = (U λ,n , p 0 ) for the corresponding object in EnrFun d (κ) and note that u κ,0 coincides with (Cat d (κ), id). For each λ ≤ λ ′ , the evident map u λ,n −→ u λ ′ ,n is a weak equivalence and each u λ,• : ∆ op −→ EnrFun d (κ) is a simplicial object where each structure map is a weak equivalence; in particular, this defines a simplicial diagram in H. Taking the (large) filtered colimit over all small cardinals λ, we then obtain a map of presheaves which induces an equivalence upon applying ψ ! . It therefore remains to verify that the target is the terminal presheaf (so that its image under ψ ! is terminal as well). Now fix an object (C, F ) ∈ H and for each λ consider the space It suffices to verify that this space is contractible for large λ; taking the (filtered) colimit over λ then shows that the target of (4.43) is terminal.
Unraveling the definitions, X λ is the classifying space of the (1, 1)-category whose objects are weak equivalences F −→ F ′ of functors C −→ Cat d , where F ′ takes λ-small values, and where a morphism is a (pointwise) trivial fibration F ′ ։ F ′′ of functors under F . For large enough λ (in particular, so that F takes λ-small values), the under-category M λ = F/Fun enr C, Cat d (λ) has the structure of a category of fibrant objects in the sense of Brown. The space X λ is then a path component in the classifying space of the wide subcategory wfM λ ⊆ M λ spanned by the trivial fibrations. By [32,Lemma 14 and 15], the classifying space of wfM λ is equivalent to that of the wide subcategory wM λ ⊆ M λ spanned by the weak equivalences. Restricting to the correct path component, this implies that for large λ, X λ is equivalent to the classifying space of the (1, 1)-category whose objects are weak equivalences F −→ F ′ of enriched functors with λ-small values, with weak equivalences F ′ −→ F ′′ as morphisms. This latter category has an initial object (F itself) and is hence contractible.
Since this is (strictly) fully faithful, the functor (4.45) is fully faithful as well. An ∞categorical proof of this can also be found in [28].   Recall that Cat d+1 has a homotopy (2, 2)-category, obtained by taking the homotopy (1, 1)-categories of all mapping (d + 1)-categories Fun d+1 (C, D). An adjunction between two (d + 1)-categories is simply an adjunction in this homotopy (2, 2)-category [42]. We then have the usual criterion for a (d + 1)-functor g : C −→ D admitting a left adjoint: For example, the map C −→ * admits a left adjoint if and only if C admits an initial object, i.e. an object ∅ such that Map C (∅, c) ≃ * for all c ∈ C.
Proof. As a preliminary observation, let us recall that any 2-functor preserves adjunctions. We are going to apply this in two situations: (a) to the 2-functor ho (2,2) and (2-)morphisms to the corresponding restriction functors and natural transformations. (b) to the localization functor Cat sPSh(∆ d−1 ) −→ ho (2,2) (Cat d ) sending each (strictly) sPSh(∆ d−1 )-enriched category to the corresponding d-category (viewed as an object in the homotopy (2, 2)-category of d-categories). Note that this is indeed a 2-functor, since it is naturally tensored over Cat (1,1) via the cartesian product.
Let us now start by assuming that g admits a left adjoint f . Applying the 2-functor (a) to this adjoint pair, we obtain an adjoint pair We In particular, the left adjoint g * (on homotopy (1, 1)-categories) is naturally isomorphic to the left derived functor of f ! : Fun enr D, sPSh(∆ d ) −→ Fun enr C, sPSh(∆ d ) . Since the left derived functor of f ! preserves representable copresheaves, it follows that g * preserves representable copresheaves as well; this is precisely the assertion that Map D (d, g(−)) is representable for all d ∈ D.
For the converse, suppose that each Map D (d, g(−)) : C −→ Cat d−1 is representable. To prove that g admits a left adjoint in ho (2,2) (Cat d ), it will suffice to find a map C op −→ D op in Cat sPSh(∆ d−1 ) which admits a (strict) right adjoint and whose image under the functor (b) is equivalent to the opposite of g : C −→ D.
To provide the desired model for g, let us choose any cofibration γ : C 0 −→ D 0 between Cat d−1 -enriched categories that models g : C −→ D. This induces a Quillen pair of (in particular) simplicial model categories Fun enr D 0 , sPSh(∆ d−1 ) : γ * given by restriction and left Kan extension. Let us now define to be the full (enriched) subcategories of all enriched functors that admit a simplicial homotopy equivalence to a corepresentable functor. Since γ ! preserves simplicial homotopy equivalences and corepresentable functors, we obtain a commuting square of categories enriched over sPSh(∆ d−1 ) where the vertical functors h are the Yoneda embeddings. By construction, both of these vertical functors are Dwyer-Kan equivalences of enriched categories, so that γ ! : C op −→ D op also maps to g op under the localization functor (b). Let us point that this would not necessarily be the case if one replaced "simplicial homotopy equivalence" by "weak equivalence" in the definition of C op (unless one imposes further fibrancy and cofibrancy conditions, which need not be preserved by both γ ! and γ * ). We claim that the restriction functor γ * also restricts to a functor γ * : D op −→ C op . Indeed, since γ * preserves simplicial homotopy equivalences, it suffices to verify that γ * sends each corepresentable functor to an object in C op . In other words, we have to show that for each d ∈ D 0 , the enriched functor Map D0 (d, γ(−)) admits a simplicial homotopy equivalence to a corepresentable object. This enriched functor is a model for Map D (d, g(−)) and is hence weakly equivalent to a corepresentable object. Since Map D0 (d, γ(−)) is projectively fibrant-cofibrant [33,Proposition A.3.3.9], it then admits a simplicial homotopy equivalence to a corepresentable as well. We conclude that γ ! : C op −→ D op admits a strict right adjoint γ * , so that its image g op in ho (2,2) (Cat d ) admits a right adjoint as well, as desired. Proof. For d ′ ∈ D ′ , let c ∈ C be a representing object for Map D (q(d ′ ), g(−)). Then (c, d ′ ) ∈ C × D D ′ provides the desired representing object for Map D ′ (d ′ , g ′ (−)). Indeed, in each of these cases consider an object (d i ) ∈ lim D i in the limit. For each i, there exists an inverse image is an equivalence. The fact that the functors C i −→ C j preserve the essential images of the fully faithful left adjoints implies that we can (inductively) lift this tuple of objects c i to an object (c i ) ∈ lim C i in the limit. This object maps to (d i ) ∈ lim D i and has the desired universal property.

4.5.
Tensoring. Let us conclude by recalling that questions about higher categories can often be reduced to questions about their underlying 1-categories when their enriched structure arises from a (co)tensoring: Definition 4.52. Let C be a (d + 1)-category and consider the (d + 1)-functor Then C is said to be tensored over Cat d if this takes values in representable functors. In this case, we will write ⊗ : Cat d × C −→ C for the induced functor.
determines a natural map µ : We will say that f is tensored over Cat d if all of these maps µ are equivalences.
Lemma 4.57. Let f : C −→ D be a tensored functor between (d + 1)-categories that are tensored over Cat d . If the underlying functor of 1-categories is an equivalence, then f is an equivalence of (d + 1)-categories as well.
Proof. Note that for any K ∈ Cat d , the map between mapping spaces in Cat d The latter is an equivalence since f is fully faithful on the underlying 1-categories, so that f is a fully faithful functor between (d + 1)-categories as well. It is essentially surjective since it is essentially surjective on the underlying 1-categories.
Proposition 4.58. Let C be a (d+1)-category that is tensored over Cat d and let F : C op −→ Cat d be a (d + 1)-functor. If F is cotensored and the underlying 1-functor is representable, then F is representable.
Proof. Let x u ∈ c 0 (F (c u )) be the element corresponding (by the Yoneda lemma) to the natural equivalence of 1-functors Map C (−, c u ) ∼ −→ c 0 (F ). Viewing x u as an element in F (c u ), the enriched Yoneda lemma provides a natural transformation of (d + 1)-functors Map C (−, c u ) −→ F . To see that this is an equivalence, note that for any K ∈ Cat d and d ∈ C, there is a commuting square The vertical functors are equivalences because F is cotensored. Since the top map induces an equivalence on underlying spaces, it follows that the bottom map induces an equivalence on the underlying spaces for all K ∈ Cat d . By the Yoneda lemma, this implies that the map x u : Map C (d, c u ) −→ F (d) is an equivalence. For instance, being (co)tensored over any full subcategory containing Θ d (and closed under cartesian products) suffices as well.

Cocartesian fibrations of higher categories
In this section we discuss the higher categorical analogues of cocartesian fibrations whose fibers are given by d-categories. We will refer to such fibrations as d-cocartesian fibrations. Most importantly, we show (Section 5.3) that d-cocartesian fibrations between (d + 1)categories form themselves the domain of a (d + 1)-cartesian fibration between (d + 2)categories Cocart d −→ Cat d+1 . 5.1. Pre-cocartesian fibrations. Let us start with the following preliminary definition, which reduces to that of a cocartesian fibration when d = 0 [1,44]: Definition 5.1. Let p : D −→ C be a map of (d + 1)-categories. Then p is said to be a pre-cocartesian fibration if the map of (d + 1)-categories is a right adjoint and the unit of the adjunction is an equivalence. A 1-morphism in D is called (p-)cocartesian if it is contained in the essential image of the left adjoint. Remark 5.3. Dually, p is said to be a pre-cartesian fibration if the map is a left adjoint and the counit of the adjunction is an equivalence. A 1-morphism in D is called (p-)cartesian if it is contained in the essential image of the right adjoint.
This notion of pre-cocartesian fibration coincides with the 2-categorical notion of an abstract cocartesian fibration from Street [47] and Riehl-Verity [44] (interpreted in the 2category of (d + 1)-categories). In particular, we have the following properties: Lemma 5.4. The class of pre-cocartesian fibrations has the following stability properties: (1) If q : E −→ D and p : D −→ C are pre-cocartesian fibrations, then pq : E −→ C is a pre-cocartesian fibration as well.
(2) If p : D −→ C is a pre-cocartesian fibration and C ′ −→ C is any map, then the base change D ′ = D × C C ′ −→ C ′ is a pre-cocartesian fibration. Finally, for (4) one can directly see that the subcategory is closed under products. It then suffices to verify that it is closed under pullbacks and limits of (countable) towers. This comes down to verifying that (5.2) admitting a fully faithful left adjoint is closed under such limits, which follows from Remark 4.51. Alternatively, one can also deduce this from Corollary 5.7, using an inductive argument to find enough cocartesian lifts in the limits. (1) For every d 2 ∈ D, the square of mapping d-categories Proof. Using the description of the mapping objects in Fun d+1 ([1], D) from Example 4.10, the map appearing in (2) can be identified with In particular, condition (1) is equivalent to condition (2) in the case where β is the identity on d 2 . Conversely, the above map arises as the base change of . It follows that (1) implies (2). In this case, the p-cocartesian morphisms are precisely the morphisms satisfying the equivalent conditions of Lemma 5.5. For the 1-categorical case, see also [1].
Proof. Using Proposition 4.49, the existence of lifts α satisfying the second condition of Lemma 5.5 is equivalent to p being a pre-cocartesian fibration.  where T and S are spaces. Then p is a pre-cocartesian fibration if and only if for any t ∈ T , the map on fibers p t : D t −→ C q(t) is a pre-cocartesian fibration. Furthermore, a morphism in D is p-cocartesian if and only if it defines a p t -cocartesian morphism in some fiber D t .
Proof. Suppose that α : d 0 −→ d 1 is an arrow in D and let τ : t 0 −→ t 1 be its image in T (which is an equivalence). For any object d 2 ∈ D, the square of (d + 1)-categories (5.6) then comes with a natural transformation to the square of spaces The horizontal maps in this square are equivalences. The square (5.6) of (d + 1)-categories (i.e. specific kinds of (d+1)-fold simplicial spaces) is then cartesian if and only if for for every τ ′ : t 1 −→ t 2 in T , the square of fibers over τ ′ is a cartesian diagram of (d + 1)-categories. This means precisely that α defines a cocartesian arrow for p t : Note that such a diagram in T is contractible, i.e. it is equivalent to a choice of basepoint t ∈ T . In other words, an arrow in D is p-cocartesian if and only if it defines a p t -cocartesian arrow where p t : D t −→ C q(t) is the map between fibers over t ∈ T . Using this and Corollary 5.7, one readily deduces that p is a pre-cocartesian fibration if and only if each p t is a pre-cocartesian fibration.

d-cocartesian fibrations.
We now turn to the definition of d-cocartesian fibrations, which proceeds by induction on d: Definition 5.10. Let p : D −→ C be a map of n-categories. We then have the following definitions, by induction on d: A strong morphism between two 0-cocartesian fibrations p and p ′ is a commuting square Dually, p is said to be a 0-cartesian fibration if p op : D op −→ C op is a 0-cocartesian fibration; strong morphisms between those are commuting squares (5.11). (b) p is said to be a d-cocartesian fibration if it satisfies the following two conditions: (1) It is a homwise (d − 1)-cartesian fibration: for every x 0 , x 1 ∈ D, the induced map between mapping objects Map D (x 0 , x 1 ) −→ Map C (p(x 0 ), p(x 1 )) is a (d − 1)cartesian fibration and the composition map determines a strong map of (d − 1)cartesian fibrations (5.12) (d) A strong morphism between d-(co)cartesian fibrations is a commuting square (5.11) such that f preserves (co)cartesian 1-morphisms and for any x 0 , x 1 ∈ D, the induced square is a strong morphism of (d − 1)-(co)cartesian fibrations.
Warning 5.13. A 0-cocartesian fibration can equivalently be viewed as a Segal copresheaf (as in Definition 4.25) whose domain is an n-category. Note that the Segal copresheaves that arise in this way form a very restrictive class, since their fibers are spaces (cf. Warning 4.26).
Remark 5.14. As usual, there are 2 d+1 variants of the notion of a d-cocartesian fibration between (d + 1)-categories, by taking opposites in the various dimensions. We will only make use of the two notions considered above, which are related by taking opposites in every dimension. Remark 5.15. The definition of a d-cocartesian fibration p : D −→ C exhibits an alternating pattern: odd-dimensional cells are required to admit p-cocartesian lifts and even-dimensional cells are required to admit p-cartesian lifts. Let us explain heuristically why this should encode a fully covariant diagram of d-categories indexed by C. First, to make sure that the fiber p −1 (c 0 ) over an object c 0 ∈ C depends covariantly on c 0 , we need to be able to associate to each 1-morphism α : c 0 −→ c 1 in C and each d 0 ∈ p −1 (c 0 ) an object α ! d 0 ∈ p −1 (c 1 ). Of course, this is done using the (essentially unique) p-cocartesian arrowα : Next, given two objects d 0 , d 1 ∈ D with images c 0 , c 1 ∈ C, consider the induced map p : Map D (d 0 , d 1 ) −→ Map C (c 0 , c 1 ). For each α : c 0 −→ c 1 in C, its fiber is given by Consequently, if α ! d 0 depends covariantly on the 1-morphism α, then the fiber p −1 (α) will depend contravariantly on α. Indeed, each 2-morphism h : α −→ β in C should give rise to a functor (5.16) h * : that precomposes with the natural transformation induced by h. In terms of the fibration p, this means that each 2-morphism h : α −→ β in C and g ∈ p −1 (β) should admit a p-cartesian lifth : h * (g) −→ g. At the level of 2-morphisms, we then obtain a functor of the form for each f ∈ p −1 (α) and g ∈ p −1 (β). For a given 2-cell h : α → β in C, postcomposition with the p-cartesian lifth induces an equivalence If the natural transformation h(d 0 ) : α ! d 0 −→ β ! d 0 depends covariantly on the 2-morphism h, then (5.16) shows that h * depends covariantly on h as well. Consequently, the fiber p −1 (h) depends covariantly on h, so that 3-morphisms should have p-cocartesian lifts. In higher dimensions, one can repeat the discussion from the last two paragraphs to arrive at the alternating list of cocartesian and cartesian conditions from Definition 5.10.
Note that for any map of n-categories p : D −→ C, the property of being a d-cocartesian fibration does not depend on whether we consider p as a map of n-categories or (n + 1)categories. Similarly, the difference between d-cocartesian and (d + 1)-cocartesian fibrations is only fiberwise: Lemma 5.17. A map p : D −→ C between n-categories is a d-cocartesian fibration if and only if it is a (d + 1)-cocartesian fibration and its fibers are d-categories.
In particular, a functor between n-categories is a d-cocartesian fibration for some d > n if and only if is an n-cocartesian fibration.
Proof. An inductive argument reduces this to the case d = 0: if p is a 0-cocartesian fibration, then certainly its fibers are spaces and it is a homwise 0-cartesian fibration (since each D(p) −→ C(p) is the base change of a map between spaces). It is a 1-cocartesian fibration since every arrow in D is p-cocartesian and every morphism in C can be lifted to an arrow in D (since p is a 0-cocartesian fibration).
Conversely, suppose that p : D −→ C is a 1-cocartesian fibration whose fibers are spaces. We have to verify that the map D(1) −→ C(1) × C({0}) D({0}) is an equivalence of (n − 1)categories. Note that this map is obtained from the map of n-categories (5.2) by taking (n−1)-cores. It particular, it admits a fully faithful left adjoint whose essential image consists of the p-cocartesian morphisms. It therefore suffices to verify that every 1-morphism in D is p-cocartesian: this follows immediately from the fact that every 1-morphism α factors as α = α ′′ • α ′ , where α ′ is cocartesian and α ′′ is a fiberwise morphism (and hence an equivalence). Remark 5.18. It will follow from straightening that the notion of a d-cocartesian fibration essentially stabilizes at the level of (d+1)-categories: if p is a d-cocartesian fibration between (d + k)-categories for k ≥ 2, then p arises as the base change of a d-cocartesian fibration D ′ −→ |C| d+1 , where |C| d+1 is the (d + 1)-category obtained by inverting all morphisms in dimension > d + 1.
Then α is said to be (p-)lateral if it is: • a p (k−1) -cocartesian 1-morphism if k is odd.
When p is an d-cartesian fibration, we define lateral k-morphisms similarly, but with the cases of k even and odd exchanged. In particular, the meaning depends on whether p is considered as a d-cocartesian or d-cartesian fibration.
Unraveling the definitions, a square (5.11) defines a strong morphism between two dcocartesian fibrations if and only if the map f preserves lateral k-morphisms for all k ≤ d. Proof. This follows by induction, using Corollary 5.9.  Using Lemma 5.21, it follows by inductive hypothesis that the vertical maps define simplicial diagrams of (d − 1)-cartesian fibrations and strong morphisms between them. Furthermore, the horizontal maps define strong morphisms between these (d − 1)-cartesian fibrations. Taking pullbacks along the rows, we then obtain another simplicial diagram of (d − 1)cartesian fibrations and strong morphisms. By Corollary 4.9, this is precisely the map q : We conclude by induction and Lemma 5.21 that q is locally (d − 1)-cartesian. From the above description one readily verifies that restriction along [1] A ′ −→ [1] A preserves lateral k-morphisms for all 1 ≤ k ≤ d.

5.3.
The (d + 2)-category of d-cocartesian fibrations. Because of Remark 5.18, we will henceforth only consider d-cocartesian fibrations between (d + 1)-categories. In light of Lemma 5.17, this is not really restrictive, since we can always choose d large enough to cover n-cocartesian fibration between m-categories for some given m and n.
Definition 5.23. We will write Cocart d ⊆ Fun d+1 [1], Cat d+1 for the sub-(d+2)-category whose objects are the d-cocartesian fibrations p : D −→ C and whose 1-morphisms are the strong morphisms between them (and all higher morphisms between those). Let us write π : Cocart d −→ Cat d+1 for the codomain projection and Cocart d (C) for the fiber over a (d + 1)-category C.
Let us furthermore write π : Cocart d −→ Cat d+1 for the induced functor between 1-cores. The fiber over a (d + 1)-category C is denoted Cocart d (C).
Remark 5.24. One can model π : Cocart d −→ Cat d+1 by a strict functor between Cat d+1 -enriched categories, where Cocart d is modeled by an enriched subcategory of Fun enr ([1], Cat d+1 ) whose objects are fibrant models for d-cocartesian fibrations and whose mapping objects are the subobjects consisting of strong morphisms.
The horizontal arrows are cocartesian since µ was a cocartesian natural transformation (Lemma 5.22). Since f (α) is a cocartesian arrow, one sees that g(α) is a cocartesian arrow as well (cf. the argument in [ To see that g preserves lateral k-morphisms for k > 1, note that for each a 0 , a 1 ∈ A there is a commuting square of mapping d-categories Map D (f a 0 , f a 1 ) Since p : D −→ C is a d-cocartesian fibration, the functor µ(a 1 ) * postcomposing with µ(a 1 ) is strong, i.e. it preserves lateral k-morphisms. Since f sends marked k-morphisms to lateral k-morphisms, it follows that µ(a 1 ) * • f does as well. On the other hand, since µ(a 0 ) is a cocartesian 1-morphism, the map µ(a 0 ) * is an equivalence of d-categories. This implies that g sends marked k-morphisms to lateral k-morphisms as well.
Proof of Proposition 5.25. Let us start by verifying that π : Cocart d −→ Cat d+1 is a homwise d-cocartesian fibration. Let p 0 : D 0 −→ C 0 and p 1 : D 1 −→ C 1 be d-cocartesian fibrations. We have to check that the induced map on mapping (d + 1)-categories is a d-cocartesian fibration. To see this, note that one can identify Next, given another d-cocartesian fibration p 2 : D 2 −→ C 2 , we have to verify that the square induced by composition gives a strong morphism of d-cocartesian fibrations. To see this, let f, g ∈ Map(p 0 , p 1 ) and f ′ , g ′ ∈ Map(p 1 , p 2 ). We will abuse notation and write f, g : D 0 −→ D 1 and f ′ , g ′ : D 1 −→ D 2 for the underlying functors, which all preserve lateral k-morphisms. Let α : f −→ g and β : f ′ −→ g ′ be lateral k-morphisms (which we depict as 1-morphisms instead of higher cells, for simplicity). Then the image of (α, β) under • is given by the horizontal composition Then f ′ α is lateral since f ′ preserves lateral k-morphisms and βg is lateral since lateral natural transformations are detected pointwise. Consequently, their horizontal composite is a lateral k-morphism as well.
It remains to check that π : Cocart d −→ Cat d+1 is a pre-cartesian fibration. Given a map f : C 1 −→ C 2 and a d-cocartesian fibration p 2 : D 2 −→ C 2 , let p 1 : D 1 = D 2 × C2 C 1 −→ C 1 be the base change. We have to show that the canonical (strong) map of cocartesian fibrations f : D 1 −→ D 2 defines a cartesian lift of f , i.e. that for any p 0 : D 0 −→ C 0 , the square is cartesian. Using (5.27), one sees that this comes down to verifying that is cartesian. This follows immediately from the fact that a functor D 0 −→ D 2 × C2 C 1 preserves lateral k-morphisms if and only if the corresponding map D 0 −→ D 2 does.

Straightening and unstraightening
Finally, we turn to straightening and unstraightening: for a small (d + 1)-category C, we will establish an equivalence between d-cocartesian fibrations over C and copresheaves C −→ Cat d . We will start by describing this as an equivalence of 1-categories (Theorem 6.1). We then use this to establish a more structured version of straightening-unstraightening that provides an equivalence of (d + 1)-categories and is furthermore (d + 2)-functorial in C (Theorem 6.20).
6.1. Straightening and unstraightening: 1-categorical version. Our first goal will be to prove the following: between (Segal) copresheaves of d-categories on C and d-cocartesian fibrations over C.
Remark 6.2. Similarly, for every (d + 1)-category C, there is an equivalence of 1-categories between cartesian fibrations and presheaves: Here the first equivalence sends a functor F : C op −→ Cat d to the functor c → F (c) (1,...,d)−op (which changes the variance with respect to higher morphisms). The second equivalence is Theorem 6.1 and the last equivalence takes (1, . . . , d + 1)-opposites to obtain a cartesian fibration (Remark 5.14), whose fibers are equivalent to the values of F .
We will prove this theorem by a repeated application of Theorem 3.1 in different simplicial directions. Concretely, the above equivalence will arise as a composition of d equivalences, whose intermediate categories consist of certain fibrations of (d + 1)-fold categories: Definition 6.3. Let C be a (d + 1)-category and let p : D −→ C be a map of (d + 1)-fold categories. By induction, we will say that p is: A strong map of 0-th stage fibrations is simply a commuting square. • a k-th stage fibration, for 1 ≤ k ≤ d, if it satisfies the following two conditions. We will write Fib (k) d ⊆ Fun [1], Cat ⊗d+1 for the subcategory whose objects are k-th stage fibrations (in particular, the codomain is a (d+ 1)-category) and whose maps are commuting squares satisfying conditions (b1) and (b2). The codomain projection defines a cartesian fibration codom : Fib Proof. Let us fix k ≥ 1 and start with the following observation: if p : D −→ C is a k-th stage fibration, then each is a (cart, left)-fibration if k is even, and a (cocart, right)-fibration if k is odd. Furthermore, each n k−1 −→ n ′ k−1 and n d−k −→ n ′ d−k induces a map preserving (co)cartesian morphisms. When k = 1 this follows from condition (a2) and the fact that C(−, 0 d ) ≃ C(−, 0, n d−1 ); for higher k it follows inductively, by repeatedly applying condition (a1).
From this point on, let us assume that k is odd; when k is even, the same argument applies up to taking opposites (in all dimensions). Note that for a (k − 1)-st stage fibration p : D −→ C, the map (6.5) is a (left, cart)-fibration and each n k−1 −→ n ′ k−1 and n d−k −→ n ′ d−k induces a strong map of (left, cart)-fibrations. Indeed, the previous paragraph shows that p is a left fibration in the k-th variable. Furthermore, setting the k-th entry equal to 0 yields a map D( n k−1 , 0, −, n d−k ) −→ C( n k−1 , 0, −, n d−k ) from a 1-category to a space. In particular, each such map is a cartesian fibration and since its cartesian arrows are the equivalences, each n k−1 −→ n ′ k−1 and n d−k −→ n ′ d−k induces a map preserving cartesian arrows.
Denoting I = ∆ ×k−1 × ∆ ×d−k op , we therefore obtain a diagram Here the functors from left to right are subcategory inclusions and the top right equivalence is the reflection functor from Theorem 3.1, applied to the k-th and (k + 1)-st variable. We will prove by induction on k that the equivalence Ψ ⊥ (k,k+1) identifies the subcategories of (k − 1)-st stage fibrations and k-th stage fibrations.
When It follows that p satisfies condition (a2) if and only if Ψ ⊥ (k,k+1) (p) does. A similar argument shows that Ψ ⊥ (k,k+1) identifies strong morphisms between (k − 1)-st stage fibrations and k-th stage fibrations, so that Ψ ⊥ (k,k+1) indeed restricts to an equivalence Fib Corollary 6.6. Consider a strong morphism of k-th stage fibrations Proof. This is evident for k = 0. For higher k, it follows from Proposition 6.4.
Lemma 6.7. Let p : D −→ C be a d-th stage fibration from a (d + 1)-fold category to a (d + 1)-category. Then the following two conditions are equivalent: (1) For any object c ∈ C, the fiber D c is a d-category in the first d-variables, and constant in the last variable. (2) D is a (d + 1)-category.
Proof. We proceed by induction on d, the case d = 0 being obvious. Note that a d-th stage fibration is either a left or a right fibration in the last variable, so that D( n d , p) ≃ D( n d , 0) × C( n d ,0) C( n d , p). This immediately shows that (2) implies (1). For the converse, note that (1) implies that the fibers of D(0) −→ C(0) are spaces, so that D(0) is a space. It then remains to verify that D(1) is a d-category. Since D(1) −→ C(1) is the opposite of a (d − 1)-st stage fibration, the inductive hypothesis says that it suffices to verify that its fibers are (d − 1)-categories. In particular, we can reduce to the case where C ≃ c 1 (C) is a 1-category.
In this case, take a morphism γ : c 0 −→ c 1 in C and consider the map on fibers D(1) γ −→ D({0}) c0 . Since the target is a space, it suffices to verify that the fiber over each point d 0 ∈ D c0 is a (d − 1)-category. Using condition (a2), we can take a cocartesian lift α : d 0 −→ d 1 of γ such that precomposition with α determines an equivalence of (d − 1)-fold categories The domain is a (d − 1)-category since the fiber D c1 is a d-category by assumption.
The left and right face of the cube are cartesian. By our assumption on p, the vertical arrows are (d − 1)-cartesian fibrations and the front and back face determine strong maps of (d − 1)-cartesian fibrations. By inductive hypothesis, this means equivalently that the vertical arrows are (d − 1)-st stage fibrations between d-categories and that the front and back face determine strong morphisms between (d−1)-st stage fibrations. Since the back and front faces are equivalent at the level of objects, Corollary 6.6 then implies that the front face is cartesian if and only if the back face is cartesian. By Corollary 5.7, this means that α is p ′ -cocartesian if and only if it is p-cocartesian. It follows that p is a pre-cocartesian fibration if and only if p ′ is a pre-cocartesian fibration, which in turn is equivalent to condition (a2), by Remark 5.8.

Morphisms.
A commuting square determines a strong morphism between d-cocartesian fibrations if and only if the induced square on d-categories of morphisms induces a strong morphism between (d − 1)-cartesian fibrations and the map c 1 (D) −→ c 1 (D ′ ) preserves cocartesian 1-morphisms. Comparing this with condition (b1) and (b2) shows inductively that such strong morphisms coincide with strong morphisms of d-th stage fibrations.
Proof (of Theorem 6.1). Composing the equivalences from Proposition 6.4 yields (6.9) that commutes with the codomain projection. Note that on both sides, the fiber over * ∈ Cat d+1 can be identified with the category Cat ⊗d of d-fold categories: in Fib (0) , a d-fold category is viewed as a (d + 1)-fold category which is constant in the first variable, and in Fib (d) it is constant in the last variable. Since the above equivalence is given by repeatedly applying the functor Ψ ⊥ , Proposition 3.15 shows that the induced equivalence on fibers over * ∈ Cat d+1 simply sends a d-fold category to itself. Now observe that there is a fully faithful inclusion coPSh Seg d ֒−→ Fib (0) , whose essential image consists of all those 0-th stage fibrations p : D −→ C with the property that for any object c ∈ C, the fiber D c is a d-category (rather than a d-fold category). The essential image of coPSh Seg d under (6.9) then consists of d-th stage fibrations whose fibers are d-categories. By Proposition 6.8, this means that (6.9) restricts to an equivalence coPSh Seg Remark 6.12. In particular, the straightening of a d-cocartesian fibration p : D −→ C between (d + 1)-categories is naturally equivalent to the straightening of p, viewed as a special kind of (d + 1)-cocartesian fibration between (d + 2)-categories.
Proof. Let φ : Cat d −→ Cat d be an equivalence of (d + 1)-categories. Since φ preserves the terminal object, we can pick a (unique) element u ∈ φ( * ). By the enriched Yoneda lemma [28, 6.2.7] (or Example 4.48), u induces a natural transformation of (d + 1)-functors u : Map Cat d ( * , −) −→ φ(−). Note that the domain can be identified with the identity on Cat d , so that u is in particular a natural transformation between two automorphisms of the 1-category Cat d . By [5,Theorem 10.1], u is a natural equivalence.
We conclude that Aut CAT d+1 (Cat d ) has an essentially unique object, given by id Cat d ≃ Map Cat d ( * , −). Using the enriched Yoneda lemma once more, the endomorphism d-category of this object can be identified with Map Cat d ( * , * ) ≃ * , which concludes the proof.
Proof of Proposition 6.10. By Theorem 6.1, the space of such equivalences is a nonempty torsor over the automorphism group G of the right fibration coPSh Seg,cart d −→ Cat d+1 . Note that this right fibration fits into a pullback square where coPSH Seg,cart d is the category of Segal copresheaves X −→ C over large (d + 1)categories, whose fibers are essentially small.
Applying Proposition 4.40 in the large setting (with κ = κ sm the supremum of all small cardinals), the right fibration π ′ corresponds under (some 1-natural version of) unstraightening to the presheaf F : CAT op d+1 −→ S big representable by the (d + 1)-category Cat d . We then obtain equivalences of spaces of automorphisms Here the first equivalence is the Yoneda embedding (4.45) and the last equivalence is unstraightening. The middle equivalence uses that CAT d+1 is a (large) ind-completion of Cat d+1 ; in particular, restriction along Cat d+1 −→ CAT d+1 is fully faithful on presheaves sending large κ sm -filtered colimits in CAT d+1 to large limits in S big (e.g. on representable presheaves). Finally, Proposition 6.13 implies that the left space is contractible.
Theorem 4.32 and Theorem 6.1 imply that the (un)straightening equivalence of Corollary 6.11 admits some extension to an equivalence of 1-categories, rather than spaces. Using this, let us record some properties of the (un)straightening at the level of objects.
Example 6.14. The proof of Theorem 6.1 shows that over C = * , the (un)straightening equivalence of Corollary 6.11 is homotopic to the identity on Cat d . Since (un)straightening intertwines restriction of copresheaves with base change of d-cocartesian fibrations, it follows that the straightening of a cocartesian fibration p : D −→ C is given pointwise by St(p) c ≃ D c . To see this, it suffices to treat the case where C = [1]. By naturality with respect to the projection [1] −→ * , the unstraightening of a constant copresheaf ∆ K : [1] −→ Cat d with value K is given by the projection π 1 : [1] × K −→ [1]. Theorem 6.1 then shows that we have a commuting diagram of mapping spaces Now note that the top zigzag is homotopic to postcomposition with γ * , while the bottom zigzag corresponds precisely to the procedure described above. The projection p : [2] lax −→ C 2 sending 0 → 0 and 1, 2 → 1 is a 1-cocartesian fibration. By Examples 6.14 and 6.15, the straightening St(p) : C 2 −→ Cat 1 is given by the diagram * Example 6.17. Given a 2-category C, let us denote by Tw lax (C) −→ C op × C the 1cocartesian fibration arising as the unstraightening of Map C : C op × C −→ Cat 1 . We will refer to Tw lax (C) as the lax twisted arrow 2-category of C.
Explicitly, Tw lax (C) is the image under the functor Ψ ⊥ of the Segal copresheaf Tw(C) −→ C op × C (Example 4.36). Unraveling the definitions, one arrives at the following explicit description of Tw lax (C) as a bisimplicial space. Note that in the definition of J[m, n], it suffices to take the colimit over the cofinal subcategory of (α, α ′ , β) such that (α, α ′ ) : [q] −→ [m] × [n] is injective and β : [p] −→ {α(q) ≤ · · · ≤ m is the inclusion of an upwards closed subset. Using this for low values of m and n, one sees that the 2-category Tw lax (C) has objects given by morphisms in C, and 1-and 2-morphisms can be depicted by diagrams in C of the form Here the left diagram defines an arrow from c 0 −→ c 1 to d 0 −→ d 1 and the right diagram defines a (blue) 2-morphism from the red to the black morphism. The projection to C op × C restricts to the left and right column.
By restricting to C op × {c} or {c} × C, we obtain an explicit description of the unstraightening of the representable presheaf Map C (−, c) and copresheaf Map C (c, −) in terms of lax versions of the over-and under-category of c. Lemma 6.18. Let p : D −→ C be a d-cocartesian fibration between (d + 1)-categories. Then the following assertions hold: (1) The induced map p ′ : c 1 D −→ c 1 C is a 1-cocartesian fibration, whose straightening St(p ′ ) can be identified with the 1-functor (2) The restriction to cocartesian arrows p ′′ : (c 1 D) cocart −→ c 1 C is a 0-cocartesian fibration, whose straightening St(p ′′ ) can be identified with the 1-functor

St(p) core0
Proof. By replacing p by its base change D × C c 1 C −→ c 1 C, we can reduce to the case where C is a 1-category. In this case, Lemma 5.17 implies that a d-cocartesian fibration q : E −→ C is a 1-cocartesian fibration if and only if E is a 1-category. Consequently, p ′ is the terminal 1-cocartesian fibration over C equipped with a natural map to p. Under straightening, this corresponds to the terminal Cat 1 -valued copresheaf on C equipped with a natural transformation to St(p). This is precisely the composite functor given in (1). Likewise, Lemma 5.17 implies p ′′ is the terminal 0-cocartesian fibration with a map to p ′ . Under straightening, this corresponds to the terminal space-valued presheaf with a natural map to St(p ′ ), which is precisely the functor given in (2). 6.3. Straightening and unstraightening: higher functoriality. Theorem 6.1 shows that for every (d + 1)-category C, there is an equivalence between the 1-category of dcocartesian fibrations over C and the 1-category underlying Fun d+1 (C, Cat d ). We will now refine this equivalence to a natural equivalence of (d + 2)-functors with values in (d + 1)categories, using Theorem 6.1 one categorical dimension higher. for the (d + 2)-functor corresponding to π under the (unique) 1-natural straightening equivalence of Corollary 6.11 (in the large setting).
We now come to our main result: Theorem 6.20 (Straightening and unstraightening). The (d + 1)-category of natural equivalences of (d + 2)-functors Cat op d+1 −→ CAT d+1 is equivalent to a single point.
Since Cat d is itself a large (d + 1)-category, it will be convenient to first study a version of the above result where we bound the cardinality. Notation 6.21. Let κ be a regular uncountable cardinal. Consider the (d + 2)-functor π : Cocart d (κ) −→ Cat d+1 whose domain is the full subcategory of Cocart d on the dcocartesian fibrations with κ-small fibers. This is a (d + 1)-cartesian fibration, and we will write Cocart κ d : Cat op d+1 −→ Cat d+1 for its image under the (natural) straightening of Corollary 6.11. Note that the values of this functor are indeed small categories: Corollary 6.11 identifies their spaces of objects with the spaces of enriched functors C −→ Cat d (κ).
Let us start by recording some properties of the (d + 2)-functor Cocart κ d . Lemma 6.22. Let f : C −→ C ′ be a functor of (d + 1)-categories. Then the (d + 1)-functor is tensored over Cat d (κ) (in particular, so are its domain and codomain).
Heuristically, this simply comes down to the fact that the tensoring of Cocart κ d (C) is given by K ⊗ (D → C) = (K × D −→ C) and that such products are stable under base change. However, since we defined the (d + 1)-functor f * somewhat abstractly, in terms the straightening from Corollary 6.11, we will provide a more detailed argument using point-set models.
Proof. The (d + 2)-functor π : Cocart d (κ) −→ Cat d+1 can be modeled by a strict functor of Cat d+1 -enriched categories π : Cocart d (κ) −→ Cat d+1 . Here Cocart d (κ) is the enriched subcategory of Fun [1], Cat d , whose objects are fibrations in Cat d+1 that present d-cocartesian fibrations with κ-small fibers and with the maximal subobject in Cat d+1 that is a model for the full sub-d-category of strong morphisms. Example 6.14 identifies the values of Cocart κ d (−) with the fibers of π. Since π is a fibration in the model category of Cat d+1 -enriched categories, the fibers of π can be modeled by the strict fibers of π. Furthermore, as we already know that the fibers Cocart κ d (C) are (d + 1)-categories (instead of (d + 2)-categories), we can model each Cocart κ d (−) by the Cat d -enriched category underlying Cocart κ d (C) = π −1 (C). As a Cat d -enriched category (rather than a Cat d+1 -enriched category), this fiber is strictly tensored over Cat d (κ): the tensoring of p : D −→ C with K ∈ Cat d (κ) is simply given by D × K −→ C. Note that for any map f : C −→ C ′ in Cat d+1 , taking (strict) base change along f defines an enriched functor f * : Cocart κ d (C ′ ) −→ Cocart κ d (C) which strictly preserves the tensoring over Cat d (κ). It therefore remains to verify that this f * indeed presents the value of Cocart κ d (−) on f . To see this, note that associated to f is a strict diagram of Cat d+1 -enriched categories Here φ is given on Cocart κ d (C ′ ) × {1} by the fiber inclusion, and on Cocart κ d (C ′ ) × {0} by the strict base change functor f * (which comes with an evident natural transformation to the fiber inclusion). One easily sees that this enriched functor φ models a map of (d+1)-cartesian fibrations (i.e. it preserves cartesian 1-morphisms). Example 6.15 then shows that the strict functor f * indeed presents (up to homotopy) the value of the straightening Cocart κ d (−) on the arrow f , so that the latter is tensored as well. Proof. By Proposition 4.58, it suffices to verify that the functor of (d + 2)-categories Cocart κ d : Cat op d+1 −→ Cat d+1 is cotensored over Cat d+1 and that the functor of 1categories c 0 Cocart κ d : Cat op d+1 −→ S is representable. We have already seen in the proof of Lemma 6.23 that this latter functor is representable by the (d + 1)-category Cat d (κ).
It remains to verify that for any K ∈ Cat d+1 and C ∈ Cat d+1 , the comparison map from Definition 4.54 is an equivalence of (d + 1)-categories. This will follow immediately from Lemma 4.57 once we show the following two claims: (a) µ induces an equivalence on the underlying 1-categories.
(b) Both the domain and codomain of µ are tensored over Cat d (κ) and µ is tensored over Cat d (κ). Let us start with assertion (a). As a first reduction, note that µ depends in particular 1-functorially on K and C. Since both its domain and codomain send colimits in K and C to limits (Lemma 6.23 and Corollary 4.9), it suffices to verify that µ is an equivalence on underlying 1-categories for generating objects K and C. We can therefore reduce to the case where K and C are contained in Θ d+1 . In particular, we may assume that K is κ-small and that for any object k ∈ K, K(k, k) ≃ * , and likewise for C.
By Theorem 4.32, Theorem 6.1 and Lemma 6.18, the 1-cartesian fibration between 1categories π : c 1 Cocart d (κ) −→ c 1 Cat d+1 arises as the localization of the homotopy cartesian fibration EnrFun d (κ) −→ Cat(Cat d ), where EnrFun d (κ) consists of enriched categories C together with an enriched functor C −→ Cat d whose values are homotopically κ-small, as in the proof of Proposition 4.40. In particular, this implies that there are equivalences c 1 Cocart κ d (K × C) ≃ c 1 Fun d+1 (K × C, Cat d (κ)) c 1 Fun d+1 K, Cocart κ d (C) ≃ c 1 Fun d+1 (K × C, Cat d (κ)) such that for any two maps K −→ K ′ and C −→ C ′ , the functoriality on the left hand side is homotopic to restriction of functors on the right hand side (we will not need any information about composition or homotopy coherence).
Using these identifications, the map µ therefore determines an endofunctor that fits into a commuting diagram of 1-categories k∈K,c∈C Cat d (κ) k∈K,c∈C Cat d (κ). Here r denotes the functor restricting to the sets of (isomorphism classes of) objects of K and C. Note that µ is (by construction) homotopic to the identity when applied to monoidal unit K = * . Up to homotopy, we may therefore assume that the bottom map in the square is the identity. The restriction functor r preserves κ-small limits and colimits and detects equivalences. By our assumption that K and C are κ-small, it has a left adjoint r ! sending each (k, c, A) to the representable functor K(k, −) × C(c, −) × A; this follows directly from the point-set presentation of r in terms of enriched functor categories. The above square then shows that µ : c 1 Fun d+1 (K × C, Cat d (κ)) −→ c 1 Fun d+1 (K × C, Cat d (κ)) preserves κ-small limits and colimits and detects equivalences as well, and admits a left adjoint µ ! . It suffices to show that for any G : K × C −→ Cat d (κ), the unit map η : G −→ µµ ! (G) is an equivalence. Since µ and µ ! both preserve κ-small colimits, it suffices to verify this when G = r ! (k, c, A) is left Kan extended from an object. By adjunction, η is then determined by a map φ k,c : A −→ K(k, k) × C(c, c) × A depending (1-)naturally on A. Since K and C were assumed to have trivial endomorphisms, we therefore obtain an endomorphism φ k,c : id −→ id of the identity functor on Cat d (κ). There is a contractible space of such natural endomorphisms by [5,Theorem 10.1].
The map φ k,c is therefore naturally (in A) homotopic to {id k } × {id c } × id A . This means precisely that η is homotopic to the identity, so in particular an equivalence. We conclude that µ ! and µ form an adjoint equivalence, so that µ does indeed induce an equivalence on the underlying 1-categories, as asserted.
We now turn to assertion (b). Note that the construction of µ (Definition 4.54) implies that for any object k ∈ K, the composite with evaluation at k is equivalent to the value of Cocart κ d (−) on the inclusion {k} × C ֒→ K × C. By Lemma 6.22, this implies that the domain and codomain of each ev k • µ are tensored over Cat d (κ) and that ev k • µ preserves the tensoring. Claim (b) will therefore follow once we know that Fun d+1 K, Cocart κ d (C) is tensored over Cat d (κ) and that evaluation at each k ∈ K preserves the tensoring.
To see this, one can either use the formalism from [28] or use the following trick. We will first prove assertion (b) in the case where C = * , so that the target of µ reduces to Fun d+1 (K, Cat d (κ)). By Example 4.56, this is indeed tensored over Cat d (κ) and restriction along {k} −→ K preserves the tensoring. Combined with (a), it follows that µ yields an equivalence Cocart κ d (K) ≃ Fun d+1 (K, Cat κ d ). Using this equivalence (with K replaced by C), the target of the map µ is then equivalent to Fun d+1 K, Cocart κ d (C) ≃ Fun d+1 K, Fun d+1 (C, Cat d (κ)) ≃ Fun d+1 K × C, Cat d (κ) . Applying Example 4.56 once more then shows that this (d + 1)-category is indeed tensored over Cat d (κ) and that evaluation at k ∈ K preserves the tensoring. We conclude that the natural transformation µ satisfies both (a) and (b), so that it is a natural equivalence as desired.
Proof of Theorem 6.20. Let us apply Theorem 6.25 in the large setting, with κ = κ sm the supremum of all small cardinals. It follows that the (d+2)-functor on large (d+1)-categories ,CAT d+1 ) F Cat d+1 , F Cat d+1 for the full sub-(d + 1)-categories spanned by the natural equivalences between (d + 2)functors. We have to show that Iso ≃ * . We have just verified that Iso is nonempty, and precomposing with any element φ ∈ Iso determines an equivalence Iso ≃ Aut(F Cat d+1 ). It therefore remains to verify that the (d + 1)-category of automorphisms of F Cat d+1 is equivalent to a point. As in Proposition 6.10, there are natural maps of (d + 1)-categories Aut CAT d+1 (Cat d ) Aut(F ) Aut(F Cat d+1 ).
The first map is an equivalence by the enriched Yoneda lemma (Section 4.4). The second map arises by restricting the natural automorphisms of F along i : Cat op d+1 ֒−→ CAT op d+1 . To see that this is an equivalence, note that together with right Kan extension, restriction along i defines a coreflective localization of (very large) (d + 2)-categories The essential image of i * consists of enriched functors CAT op d+1 −→ CAT d+1 whose underlying 1-functor preserves large, κ sm -cofiltered limits; this can be verified by presenting everything by enriched (model) categories of strictly enriched functors. In particular, the representable functor F is contained in the essential image of i * , so that restriction along i induces an equivalence on its automorphism (d + 1)-categories. The result now follows since Aut CAT d+1 (Cat d ) ≃ * by Proposition 6.13.

Appendix A. Technicalities on categories of fibrant objects
The purpose of this appendix is to describe a version of the Grothendieck construction for categories of fibrant objects, in the following sense: Definition A.1. By a category of fibrant objects M we will mean a (1, 1)-category with two classes of maps, called weak equivalences and fibrations, both containing the isomorphisms and closed under composition, such that the following conditions hold: (1) The weak equivalences have the 2-out-of-6 property.
(2) M admits pullbacks along fibrations and the base change of an (acyclic) fibration is again an (acyclic) fibration.
(3) M admits a terminal object and each x −→ * is a fibration.
(4) Each map in M admits a factorization into a weak equivalence, followed by a fibration.
A map between categories of fibrant objects f : M −→ N is a functor preserving fibrations, weak equivalences, the terminal object and pullbacks along fibrations. Let us write FibCat for the (2, 1)-category of categories of fibrant objects.
Our condition that weak equivalences have the 2-out-of-6 property is slightly nonstandard; it is equivalent to the assertion that a morphism in M is a weak equivalence if and only if its image in the localization M[W −1 ] is an equivalence (see [38,Theorem 7.2.7]  exists a factorization f = pi with p : b ′ 0 −→ b 1 a fibration and i : b 0 −→ b ′ 0 a weak equivalence, together with a factorization g = i * (q)j with q : z ։ p * y a fibration in E(b ′ 0 ) and j : x −→ i * z a weak equivalence in E(b 0 ). Define a map x 0 −→ x 1 in E to be a weak equivalence (fibration) if it factors (uniquely) as x 0 −→ x −→ x 1 where the second map is a p-cartesian lift of a weak equivalence (fibration) and the first map is a weak equivalence (fibration) in E(px 0 ). This makes p : E −→ B a map between categories of fibrant objects.
Proof. One easily verifies that pullbacks along fibrations exist and that the base change of an (acyclic) fibration remains an (acyclic) fibration. The terminal object in E is given by * ∈ E( * ) and each x −→ * is a fibration. Condition (a) implies that for any weak equivalence f : b 0 −→ b 1 , the functor f * : E(b 1 ) −→ E(b 0 ) detects weak equivalences (which are detected in the localization). Using this, one readily verifies that the weak equivalences have the 2-out-of-6 property. Finally, the factorization condition is precisely (b). Indeed, for a map g : x −→ f * y in E(b 0 ), we can consider the adjoint map i ! x −→ p * y and factor it into i ! x ∼ −→ z followed by a fibration q : z ։ p * y. One then obtains the desired factorization g = i * (q)j, where j : x −→ i * i ! x −→ i * z is a model for the derived unit map and hence an equivalence by condition (a).
(0) objects are the objects of M.
(1) mapping categories Span M (x 0 , x 1 ) are the full subcategories of spans x 0 ←− x 01 −→ x 1 where the left morphism α : x 01 −→ x 0 is an acyclic fibration and β : x 01 −→ x 1 is any morphism (and composition by composition of spans). We will abbreviate such a span by (α, x 01 , β) : x 0 x 1 . Taking classifying spaces then determines a 1-categorical algebra (i.e. a Segal space) whose completion models the localization [29]. In fact, this description works for any relative category (C, W ) where W is generated under the 2-out-of-3 property by a class of 'acyclic fibrations' which is stable under base change.
Proof. Let us start by considering the map of (2, 2)-categories p : Span E −→ Span B . For each x 0 , x 1 ∈ E, the induced functor on mapping (1, 1)-categories is a cartesian fibration, whose fiber over a span (α, b, β) : px 0 px 1 can be identified with Span E(b) (α * x 0 , β * x 0 ). Using condition (a) and Remark A.8, it follows that for each morphism γ : b ′ ∼ −→ b in Span B (px 0 , px 1 ), the induced map on fibers induces an equivalence on classifying spaces. It follows that (A.9) satisfies the conditions of Quillen's Theorem B; consequently, the classifying space functor | − | : Cat (1,1) −→ S preserves all pullbacks along it.
In particular, this implies that the pullback square of (2, 2)-categories induces a (homotopy) pullback square of Segal spaces upon taking classifying spaces of all mapping categories. To prove assertion (1), we have to show that this also induces a homotopy pullback square on completions. This follows as soon as Span E −→ Span B induces an isofibration on homotopy (1, 1)-categories (cf. [19,Corollary 4.4]). To see this, note that by the 2-out-of-6 property, any isomorphism in ho(Span B ) arises from a span (α, b, β) : b 0 b 1 in B with α a trivial fibration and β a weak equivalence. It therefore suffices to prove the following stronger assertion: ( * ) For any span (α, b, β) : b 0 b 1 in B with α a trivial fibration and any x 1 ∈ E b1 , there exists a lift to a span (α, x 12 ,β) : x 0 x 1 withα a trivial fibration andβ homotopy cartesian.
To see this, first take β * x 1 ∈ E b and recall that α * : E b0 −→ E b induces an equivalence on localizations. This implies that there exists some x 0 ∈ E b0 together with a zig-zag This proves (1). For (2) and (3), note that by Remark A.6 and Remark A.8, the localization of E hocart arises from the (2, 2)-category Span E hocart . Since being a homotopy p-cartesian morphism is invariant under weak equivalences, Span E hocart (x 0 , x 1 ) −→ Span E (x 0 , x 1 ) is an inclusion of connected components, so that E hocart [W −1 ] −→ E[W −1 ] is indeed the inclusion of a wide subcategory. Assertions (2) and (3) then follow from ( * ) as soon as every arrow in E hocart [W −1 ] is a cartesian arrow.
Let (α, x 12 ,β) : x 1 x 2 be a morphism in Span E hocart , so thatβ is homotopy p-cartesian, and let (α, b 12 , β) : b 1 b 2 denote its image in B. To see that this map is an equivalence, we have to show that for any x 0 ∈ E with image b 0 = px 0 in B, the square of categories (A.10) Recall from the beginning of the proof that the fibers of the vertical maps in (A.10) are themselves given by categories of spans. Unraveling the definition, the map between the vertical fibers over (γ, b 01 , δ) can then be identified with the composite The first map is an equivalence since (α ′ ) * induces an equivalence on localizations. The second map takes the postcomposition with the span (δ ′ ) * α * x 1 ←− (δ ′ ) * x 12 −→ (δ ′ ) * β * x 2 , where both maps are weak equivalences sinceα andβ were homotopy cartesian. This map induces an equivalence on classifying spaces as well. Finally, for (4), let us start by recalling that for any cartesian fibration p : D −→ C of 1-categories, an arrow in D is p-cocartesian if and only if it is locally p-cocartesian, i.e. cocartesian in the base change of p to a 1-simplex [33,Corollary 5.2.2.4]. In the present situation, the observations at the beginning of the proof imply that the base change of E[W −1 ] −→ B[W −1 ] to the 1-simplex f : b 0 −→ b 1 can be computed at the level of (2, 2)categories. Unraveling the definitions as in the proofs of parts (2) and (3), we reduce to showing that for any x 2 ∈ E(b 1 ), induces an equivalence on classifying spaces. Here the second map precomposes spans with the unit η : x 0 −→ f * f ! x 0 . Explicitly, (A.11) sends f ! x 0 ←− x 12 −→ x 2 to the span This induces an equivalence on classifying spaces since it admits a left adjoint, sending