Proof. We first observe that for any object $T\in {\mathcal {O}}^{\otimes }$ , the induced morphism on fibers $\mathbf {D}^{\otimes }_T\to ~\mathbf {E}^{\otimes }_T$ is a coCartesian fibration. Indeed, if $T\in {\mathcal {O}}^{\otimes }_{\langle 1\rangle }$ , this is our assumption. Now, if $T\in {\mathcal {O}}^{\otimes }_{\langle n\rangle }$ , then using the notation from [Reference Lurie17, Remark 2.1.1.15], we can write $T= T_1\oplus ... \oplus T_n$ with $T_i\in {\mathcal {O}}^{\otimes }_{\langle 1\rangle }$ , and by the definition of ${\mathcal {O}}$ -monoidal category, we have $\mathbf {D}^{\otimes }_T\simeq \prod _{i=1}^n \mathbf {D}^{\otimes }_{T_i}$ , and similarly for $\mathbf {E}^{\otimes }$ , in a compatible way. Because products of coCartesian fibrations are coCartesian, this proves the claim.

We are now in the situation of Lemma 2.9: the assumption that ${\mathcal {O}}$ -operations preserve coCartesian edges is exactly the assumption that $e_! : X_s\to X_t$ sends $p_s$ -coCartesian edges to $p_t$ -coCartesian edges (cf. Remark 2.6), in the notation of that lemma.

Proof. First note that $\mathrm {pr}:\mathbf {C}\times _{\mathbf {E}}\mathbf {D}\to \mathbf {C}$ is a coCartesian fibration, as it is pulled back from one. Therefore, the composite $\mathbf {C}\times _{\mathbf {E}}\mathbf {D}\to \mathbf {C}\to S$ is also a coCartesian fibration.

The claim about coCartesian edges follows from [Reference Lurie16, Proposition 2.4.1.3(3)]: indeed, edges that are $\mathrm {pr}$ -coCartesian lying over edges in $\mathbf {C}$ that are p-coCartesian are themselves $\pi $ -coCartesian, and because $\mathbf {C}\times _{\mathbf {E}}\mathbf {D}\to \mathbf {C}$ and $\mathbf {C}\to S$ are coCartesian fibrations, there is a sufficient supply of those, hence they are the only $\pi $ -coCartesian edges in $\mathbf {C}\times _{\mathbf {E}}\mathbf {D}$ . In particular, they map to p-coCartesian edges in $\mathbf {C}$ .

Proof of Proposition 2.4

By Lemma 2.10, $\mathbf {D}^{\otimes }\to \mathbf {E}^{\otimes }$ is a coCartesian fibration.

Therefore, by Lemma 2.12 and the assumption that $\mathbf {D}^{\otimes }\to \mathbf {E}^{\otimes }$ is a morphism of ${\mathcal {O}}$ -monoidal categories, it follows that $\mathbf {C}^{\otimes }\times _{\mathbf {E}^{\otimes }}\mathbf {D}^{\otimes }\to {\mathcal {O}}^{\otimes }$ is a coCartesian fibration, and that the projection to $\mathbf {C}^{\otimes }$ preserves coCartesian edges. To simplify notation, we will call the pullback $K^{\otimes }$ .

According to [Reference Lurie17, Definition 2.1.2.13], to conclude we now have to show that for every $T\in {\mathcal {O}}^{\otimes }$ , decomposing as $T= T_1\oplus ... \oplus T_n$ , the inert morphisms $T\to T_i$ induce an equivalence $K^{\otimes }_T \simeq \prod _{i=1}^n K^{\otimes }_{T_i}$ .

But this follows from the same claim for $\mathbf {C}^{\otimes },\mathbf {D}^{\otimes },\mathbf {E}^{\otimes }$ and the fact that $\mathbf {C}^{\otimes }\to \mathbf {E}^{\otimes }$ preserves coCartesian edges over inert edges (even if it does not preserve all coCartesian edges).

Furthermore, Lemma 2.12 also shows that $K^{\otimes }\to \mathbf {C}^{\otimes }$ preserves coCartesian edges, so in particular, it is a morphism of ${\mathcal {O}}$ -monoidal categories.

Sketch of proof of Theorem 3.1.

By [Reference Lurie17, Proposition 2.4.1.7], lax ${\mathcal {O}}$ -monoidal functors $\mathbf {C}\to \mathbf {Cat} $ are equivalently functors $\mathbf {C}^{\otimes }\to \mathbf {Cat} $ (as opposed to $\mathbf {C}^{\otimes }\to \mathbf {Cat} ^{\times }$ ) satisfying an extra condition, namely that of being a “lax Cartesian structure” in the sense of [Reference Lurie17, Definition 2.4.1.1].

So we find equivalences

$$ \begin{align*}\operatorname{Fun}^{lax-{\mathcal{O}}}(\mathbf{C},\mathbf{Cat} )\simeq \operatorname{Fun}^{lax-cart}(\mathbf{C}^{\otimes},\mathbf{Cat} )\subset \operatorname{Fun}(\mathbf{C}^{\otimes},\mathbf{Cat} )\simeq \mathbf{coCart}_{\mathbf{C}^{\otimes,}}\end{align*} $$

where the last equivalence is the unstraightening equivalence.

We claim two things about the composite fully faithful embedding: firstly, that its image is exactly the category of ${\mathcal {O}}$ -monoidally coCartesian fibrations $\mathbf {coCart}_{\mathbf {C}}^{\mathcal {O}}\subset \mathbf {coCart}_{\mathbf {C}^{\otimes }}$ (which will conclude the proof); and second that on objects it can be described as

with the ${\mathcal {O}}$ -monoidal structure described earlier.

The first verification amounts to the verification that $\mathbf {coCart}_{\mathbf {C}}^{\mathcal {O}}$ is indeed equivalently the category of $\mathbf {C}^{\otimes }$ -monoidal categories in the sense of [Reference Lurie17, Definition 2.1.2.13] (which will rely on [Reference Lurie17, Proposition 2.1.2.12]), and is done by Hinich in [Reference Hinich10].

The second verification is strictly speaking not needed for the proof but we make it nonetheless to connect with the pullback description from above, for intuition and concreteness. Essentially by design, the construction described in the previous paragraph fits in a pullback square:

while the one coming from the above string of equivalences fits in a pullback square:

Thus to compare the two, it suffices to construct a pullback square:

We now observe that the construction of the map $D^{\times }\to D$ from [Reference Lurie17, Proposition 2.4.1.5] is natural in the category with finite products D, so that we have such a naturality square. Thus, we have a map from to the pullback over $\mathbf {Cat} ^{\times }$ , encoding a map of symmetric monoidal coCartesian fibrations over $\mathbf {Cat} $ , which clearly induces an equivalence on underlying categories, and is thus an equivalence as well.

Observation 4.3. $\mathbf {LFib}^{rep}\subset \operatorname {Fun}(\Delta ^1,\mathbf {Cat} )$ is closed under (Cartesian) products, and the functor $ev_1$ preserves those. It is therefore canonically symmetric monoidal for the Cartesian symmetric monoidal structure.

Furthermore, $ev_1$ -coCartesian edges are preserved under Cartesian products, as is easily seen from their description above.

Proof. See the discussion in [Reference Lurie17, Corollaries 2.4.1.8 and 2.4.1.9].

Proof. As pullbacks of coCartesian fibrations are coCartesian fibrations, it is clear that $\mathbf {coCart}\subset \operatorname {Fun}(\Delta ^1,\mathbf {Cat} )$ is a sub-Cartesian fibration.

In particular, it is a Cartesian fibration. To prove that it is also coCartesian, it therefore suffices to prove that the pullback functors have left adjoints ([Reference Lurie16, Corollary 5.2.2.5]).

If $f: \mathbf {C}\to \mathbf {D}$ is a functor, then the pullback functor $\mathbf {coCart}_{\mathbf {D}}\to \mathbf {coCart}_{\mathbf {C}}$ gets identified, under the straightening/unstraightening equivalence, with $f^* :~\operatorname {Fun}(\mathbf {D},\mathbf {Cat} )\to \operatorname {Fun}(\mathbf {C},\mathbf {Cat} )$ , which has a left adjoint given by left Kan extension. This proves the claim.

Proof. This can be formulated as saying that for any pair of functors $\alpha : \mathbf {C}_0\to \mathbf {C}_1, \beta :~\mathbf {D}_0\to ~\mathbf {D}_1$ , the following natural transformation is an equivalence:

that is, the square strictly commutes, instead of only commuting up to a non-invertible natural transformation (this natural transformation comes from the fact that

$$ \begin{align*}\mathbf{coCart}\times\mathbf{coCart}\xrightarrow{\times}\mathbf{coCart}\times_{\mathbf{Cat} }(\mathbf{Cat} \times\mathbf{Cat} )\end{align*} $$

is a morphism over $\mathbf {Cat} \times \mathbf {Cat} $ , between coCartesian fibrations; and so we get canonical lax commutative squares as above).

Under the straightening/unstraightening equivalence, the functor $\mathbf {coCart}_{\mathbf {C}_0}\times ~\mathbf {coCart}_{\mathbf {D}_0} \to ~\mathbf {coCart}_{\mathbf {C}_0\times \mathbf {D}_0}$ gets identified with $\operatorname {Fun}(\mathbf {C}_0,\mathbf {Cat} )\times \operatorname {Fun}(\mathbf {D}_0,\mathbf {Cat} )\to \operatorname {Fun}(\mathbf {C}_0\times \mathbf {D}_0,\mathbf {Cat} )$ given by $(F,G)\mapsto ~((c,d)\mapsto ~F(c)\times ~G(d))$ , or more precisely, $\Pi \circ p_{\mathbf {C}_0}^*F \times p_{\mathbf {D}_0}^*G$ , where $p_{\mathbf {C}_0}$ (resp., $p_{\mathbf {D}_0}$ ) is the projection from the product $\mathbf {C}_0\times \mathbf {D}_0\to \mathbf {C}_0$ (resp., $\mathbf {D}_0$ ) and $\Pi :\mathbf {Cat} \times \mathbf {Cat} \to \mathbf {Cat} $ is the product functor.

Indeed, if $X\to \mathbf {C}_0, Y\to \mathbf {D}_0$ are coCartesian fibrations, there is an equivalence of coCartesian fibrations over $\mathbf {C}_0\times \mathbf {D}_0$ (natural in $X,Y$ ) $X\times Y\simeq p_{\mathbf {C}_0}^*X\times _{\mathbf {C}_0\times \mathbf {D}_0} p_{\mathbf {D}_0}^*Y$ , this fiber product is the Cartesian product in $\mathbf {coCart}_{\mathbf {C}_0\times \mathbf {D}_0}$ , and finally restriction in functor categories corresponds to pullback in fibrations.

We denote this abusively by $F\times G$ , hoping that no confusion arises.

One checks similarly that the canonical natural transformation from above, in terms of functors, and evaluated at $(F,G)$ is the canonical morphism $(\alpha \times \beta )_!(F\times G)\to \alpha _! F\times \beta _! G$ .

But this is an equivalence, because left Kan extensions are pointwise ( $\mathbf {Cat} $ is cocomplete) so that the canonical morphism from above is pointwise of the form:

$$ \begin{align*}(\mathrm{colim}_{ (\mathbf{C}_0\times\mathbf{D}_0)\times_{\mathbf{C}_1\times\mathbf{D}_1}(\mathbf{C}_1\times\mathbf{D}_1)_{/(c,d)}} F\times G) \to (\mathrm{colim}_{ (\mathbf{C}_0\times_{\mathbf{C}_1}(\mathbf{C}_1)_{/c} )} F)\times (\mathrm{colim}_{ (\mathbf{D}_0\times_{\mathbf{D}_1}(\mathbf{D}_1)_{/d} )} G)\end{align*} $$

and the comma categories appearing in the indexing of these colimits are equivalent: the canonical functor

$$ \begin{align*}(\mathbf{C}_0\times\mathbf{D}_0)\times_{\mathbf{C}_1\times\mathbf{D}_1}(\mathbf{C}_1\times\mathbf{D}_1)_{/(c,d)}\to (\mathbf{C}_0\times_{\mathbf{C}_1}(\mathbf{C}_1)_{/c} )\times (\mathbf{D}_0\times_{\mathbf{D}_1}(\mathbf{D}_1)_{/d} )\end{align*} $$

is an equivalence—here, we use that for any $C\in \mathbf {Cat} $ , $C\times -$ commutes with colimits to reduce to a comparison between indexing categories. This concludes the proof.

Proof. The claim about products is obvious.

For $ev_1$ -coCartesian morphisms, we again use straightening: by looking at fibers, the claim is that for any $f: \mathbf {C}\to \mathbf {D}$ , the natural transformation in the following square is an equivalence:

Under straightening, this square gets identified with:

and the canonical transformation being invertible is simply a witness of the naturality of the Yoneda embedding (and the fact that the inclusion $\mathbf {An}\subset \mathbf {Cat} $ preserves colimits).

Proof. Both $\mathbf {LFib}^{rep}$ and $\mathbf {coCart}$ have finite products, and the functor $ev_1$ preserves products for both, so we can see it as a symmetric monoidal functor to $\mathbf {Cat} $ , where both the domain and codomain are viewed as Cartesian symmetric monoidal categories.

Furthermore, we have seen that coCartesian edges are closed under product, so that they are both symmetric monoidally coCartesian fibrations over $\mathbf {Cat} $ .

Theorem 3.1 implies (up to passing to a larger universe) that their straightening is canonically lax symmetric monoidal, and the transformation between them too.

Proof of Lemma 4.14

The second claim about the fiber is immediate from the second construction, so we focus on the first part of the lemma.

The observation here is that the lax ${\mathcal {O}}$ -monoidal functor $\mathbf {F}^{\otimes }\to \mathbf {Cat} ^{\times }$ associated with $\mathbf {E}^{\otimes }\to \mathbf {F}^{\otimes }$ fits in a pullback square

so in particular, we have a bigger pullback diagram

The top left-hand corner is the $\mathbf {E}_f^{\otimes }$ from the first construction, while the bottom horizontal composite by definition classifies $\mathbf {E}_f$ as an ${\mathcal {O}}'$ -algebra in $\mathbf {Cat} $ via the lax symmetric monoidal functor $\mathbf {F}^{\otimes }\to \mathbf {Cat} ^{\times }$ .

The equivalence between ${\mathcal {O}}'$ -algebras in $\mathbf {Cat} $ and ${\mathcal {O}}'$ -monoidal categories now precisely tells us that $\mathbf {E}_f^{\otimes }$ is the ${\mathcal {O}}'$ -monoidal category corresponding to the ${\mathcal {O}}'$ -algebra $\mathbf {E}_f$ in $\mathbf {Cat} $ .

Proof of Proposition 4.12

We begin by proving the statement replacing $\mathbf {Pr}^{\mathrm {L}}$ with $\widehat {\mathbf {Cat} }$ .

For $\widehat {\mathbf {Cat} }$ , we observe that $\mathbf {C}\mapsto \mathbf {coCart}_{\mathbf {C}}$ , as a functor $\mathbf {Cat} \to \widehat {\mathbf {Cat} }$ , is lax symmetric monoidal, and therefore induces a canonical lax symmetric monoidal functor $\mathbf {Cat} \to ~\mathbf {Mod}_{\mathbf {coCart}_*}(\widehat {\mathbf {Cat} })$ , because $*$ is the unit in $\mathbf {Cat} $ .

We are therefore left with two verifications: firstly, that the equivalence $\mathbf {coCart}_* \simeq \mathbf {Cat} $ can be made symmetric monoidal, that is, that $\mathbf {coCart}_*$ is Cartesian monoidal; and secondly, the claim about the pointwise $\mathbf {Cat} $ -linear structure on $\mathbf {coCart}_{\mathbf {C}}$ .

For the first part, we apply Lemma 4.14 to ${\mathcal {O}}={\mathcal {O}}'= \mathrm {Comm}, \mathbf {E}^{\otimes }= \mathbf {coCart}^{\times }, \mathbf {F}= \mathbf {Cat} ^{\times }$ , from which it follows that the monoidal structure on $\mathbf {coCart}_*$ is Cartesian, and hence, the correct one.

For the second claim, we again use Lemma 4.14, with ${\mathcal {O}}= \mathrm {Comm}$ and ${\mathcal {O}}'=$ the operad classifying left modules, together with the observation that we have a map $\mathbf {coCart}^{\times }\to ~\operatorname {Fun}(\Delta ^1,\mathbf {Cat} )^{\times }$ over $\mathbf {Cat} ^{\times }$ . We then only have to check that the map we obtain on fibers over $({\mathcal {O}}')^{\otimes }\xrightarrow {(*,\mathbf {C})}~\mathbf {Cat} ^{\times }$ , which is a priori a map of ${\mathcal {O}}'$ -operads, is a map of ${\mathcal {O}}'$ -monoidal categories—but this follows from the fact that $\mathbf {coCart}_{\mathbf {C}} \subset \mathbf {Cat} _{/\mathbf {C}}$ is stable under $-\times K$ for $K\in \mathbf {Cat} $ .

We now explain how to replace $\widehat {\mathbf {Cat} }$ with $\mathbf {Pr}^{\mathrm {L}}$ : the Lurie tensor product by definition witnesses $(\mathbf {Pr}^{\mathrm {L}})^{\otimes }$ as a (non-full) sub-operad of $\widehat {\mathbf {Cat} }^{\times }$ , so moving from the latter to the former is just a property, of the objects involved (namely, that of being presentable) and the functors involved (preserving colimits in each variable). In our situation, the objects are $\mathbf {coCart}_{\mathbf {C}}$ for some $\mathbf {C} \in \mathbf {Cat} $ , and these are indeed presentable, and the morphisms are either $\mathbf {Cat} \times \mathbf {coCart}_{\mathbf {C}}\to \mathbf {coCart}_{\mathbf {C}}$ (for the $\mathbf {Cat} $ -module structure) or $\mathbf {coCart}_{\mathbf {C}}\times ~\mathbf {coCart}_{\mathbf {D}} \to ~\mathbf {coCart}_{\mathbf {C}\times \mathbf {D}}$ (for the lax symmetric monoidal structure). To verify that these do preserve colimits in each variable, one can either use un/straightening, or the results from Appendix A. Let us indicate how the latter verification works: Proposition A.1 allows us to reduce to the same statements for $\mathbf {Cat} _{/\mathbf {C}}, \mathbf {Cat} _{/\mathbf {D}}$ . For these, one may use that colimits in slice categories are underlying, so we are reduced to the same statement for $\mathbf {Cat} $ , but now for any $E\in \mathbf {Cat} , E\times -$ has a right adjoint and hence preserves colimits.

All in all, we get a lift to $\mathbf {Mod}_{\mathbf {Cat}} (\mathbf {Pr}^{\mathrm {L}})$ .

Observation 4.17. The forgetful functor $\mathbf {Mod}_{\mathbf {Cat}} (\mathbf {Pr}^{\mathrm {L}})\to \mathbf {Pr}^{\mathrm {L}}\to \widehat {\mathbf {Cat} }$ is canonically lax symmetric monoidal, so we can also view $\mathbf {C}\mapsto \operatorname {Fun}(\mathbf {C},\mathbf {Cat} )$ as a lax symmetric monoidal functor $\mathbf {Cat} \to \widehat {\mathbf {Cat} }$ .

Proof. This follows essentially from Corollary 4.11, the universal property of presheaf-categories, and the fact that $\operatorname {Fun}(\mathbf {C},\mathbf {An})\otimes \mathbf {Cat} \simeq \operatorname {Fun}(\mathbf {C},\mathbf {Cat} )$ .

In more detail, we have just explained how $\mathbf {C}\mapsto \mathbf {coCart}_{\mathbf {C}}$ lifts to a symmetric monoidal functor $\mathbf {Cat} \to \mathbf {Mod}_{\mathbf {Cat}} (\mathbf {Pr}^{\mathrm {L}})$ , and the forgetful functor $\mathbf {Pr}^{\mathrm {L}}\to \widehat {\mathbf {Cat} }$ has a partial symmetric monoidal left adjoint defined on small categories, and given there by $\operatorname {Fun}((-)^{\mathrm {op}},\mathbf {An})$ .

$\mathbf {Mod}_{\mathbf {Cat}} (\mathbf {Pr}^{\mathrm {L}})\to \mathbf {Pr}^{\mathrm {L}}$ has a symmetric monoidal left adjoint given by $\mathbf {Cat} \otimes -$ , and $\operatorname {Fun}((-)^{\mathrm {op}},\mathbf {An})\otimes \mathbf {Cat} \simeq \operatorname {Fun}((-)^{\mathrm {op}},\mathbf {Cat} )$ , symmetric monoidally by definition.

It therefore follows that the symmetric monoidal transformation $\mathbf {C}^{\mathrm {op}}\to ~\mathbf {coCart}_{\mathbf {C}}$ from Corollary 4.11 has a unique $\mathbf {Cat} $ -linear colimit-preserving extension $\operatorname {Fun}(\mathbf {C},\mathbf {Cat} )\to \mathbf {coCart}_{\mathbf {C}}$ , which has a canonical $\mathbf {Cat} $ -linear symmetric monoidal structure.

It is now easy to see that the underlying functor $\operatorname {Fun}(\mathbf {C},\mathbf {Cat} )\to \mathbf {coCart}_{\mathbf {C}}$ is just the unstraightening equivalence: it is a colimit-preserving $\mathbf {Cat} $ -linear functor $\operatorname {Fun}(\mathbf {C},\mathbf {Cat} )\to ~\mathbf {coCart}_{\mathbf {C}}$ whose restriction along the Yoneda embedding $\mathbf {C}^{\mathrm {op}}\to \operatorname {Fun}(\mathbf {C},\mathbf {An})\to \operatorname {Fun}(\mathbf {C},\mathbf {Cat} )$ agrees with the restriction of the unstraightening equivalence. The unstraightening equivalence also has these properties (it is an equivalence, so it preserves colimits, and it is $\mathbf {Cat} $ -linear—this is folklore, but see Lemma A.6 for a proof), and this implies that they are equivalent by the universal property of presheaves and the equivalence $\operatorname {Fun}(\mathbf {C},\mathbf {Cat} )\simeq \operatorname {Fun}(\mathbf {C},\mathbf {An})\otimes \mathbf {Cat} $ .

Because equivalences of symmetric monoidal functors are pointwise, that is, the evaluation functors at $\mathbf {C}, \mathbf {C}\in \mathbf {Cat} $ , are jointly conservative as functors $\operatorname {Fun}^{\otimes }(\mathbf {Cat} , \mathbf {Mod}_{\mathbf {Cat}} (\mathbf {Pr}^{\mathrm {L}}))\to ~\widehat {\mathbf {Cat} }$ , the rest of the claim follows.

Proof. Because both this structure and the Day convolution structure are presentably $\mathbf {Cat} $ -linearly ${\mathcal {O}}$ -monoidal, it suffices to prove that the Yoneda embedding $\mathbf {C}^{\mathrm {op}}\to ~\operatorname {Fun}(\mathbf {C},\mathbf {Cat} )$ is ${\mathcal {O}}$ -monoidal, where the target has Day convolution and the source has the monoidal structure coming from the symmetric monoidality of $\mathbf {Cat} \to ~\mathbf {Cat} , \mathbf {D}\mapsto ~\mathbf {D}^{\mathrm {op}}$ .

But this symmetric monoidality comes from the fact that this functor preserves products (cf. Lemma 4.4). In particular, the structure on $\mathbf {C}^{\mathrm {op}}$ is just the usual $^{\mathrm {op}}$ of ${\mathcal {O}}$ -monoidal structures, and the claim boils down to the claim that for an ${\mathcal {O}}$ -monoidal category, the Yoneda embedding $\mathbf {C}\to \operatorname {Fun}(\mathbf {C}^{\mathrm {op}},\mathbf {An})$ is ${\mathcal {O}}$ -monoidal.

This is [Reference Lurie17, Proposition 4.8.1.10]—more specifically, the variant of [Reference Lurie17, Proposition 4.8.1.12], but with a more general operad than $\mathrm {Comm}$ .

Proof of Lemma 5.2

By definition, Theorem A induces an ${\mathcal {O}}$ -monoidal structure on $\mathbf {coCart}_{\mathbf {C}}$ in the following way: $\mathbf {C}$ is an ${\mathcal {O}}$ -algebra in $\mathbf {Cat} $ , and $\mathbf {coCart}_\bullet : \mathbf {Cat} \to \widehat {\mathbf {Cat} }$ is obtained from Theorem 3.1 by straightening the symmetric monoidally coCartesian fibration $\mathbf {coCart}\to \mathbf {Cat} $ .

Up to passing to a bigger universe, we are therefore exactly in the situation of Lemma 4.14. In particular, we have a pullback square of the form

Note that both $\mathbf {coCart},\mathbf {Cat} $ are Cartesian monoidal.

We similarly have, by applying the same argument to $\operatorname {Fun}(\Delta ^1,\mathbf {Cat} )$ (which is easily seen to be Cartesian monoidal) in place of $\mathbf {coCart}$ , a pullback square:

Recall from Remark 4.6 that the map $\mathbf {coCart}\to \operatorname {Fun}(\Delta ^1,\mathbf {Cat} )$ is not a morphism of coCartesian fibrations, but it is nonetheless a morphism over $\mathbf {Cat} $ (in fact, it is a morphism of Cartesian fibrations over $\mathbf {Cat} $ ) so that we also have a commutative triangle:

Because $\mathbf {coCart}\subset \operatorname {Fun}(\Delta ^1,\mathbf {Cat} )$ preserves products and is a non-full subcategory, $\mathbf {coCart}^{\times }\subset \operatorname {Fun}(\Delta ^1,\mathbf {Cat} )^{\times }$ is a non-full suboperad.

Explicitly, given coCartesian fibrations $\mathbf {D}_1\to \mathbf {C}_1, ..., \mathbf {D}_n\to \mathbf {C}_n, \mathbf {D}_\infty \to \mathbf {C}_\infty $ , which we will abusively denote by only referring to their total category, the inclusion

$$ \begin{align*}\operatorname{map}_{\mathbf{coCart}^{\times}}(\mathbf{D}_1,...,\mathbf{D}_n; \mathbf{D}_\infty)\subset \operatorname{map}_{\operatorname{Fun}(\Delta^1,\mathbf{Cat} )^{\times}}(\mathbf{D}_1,...,\mathbf{D}_n; \mathbf{D}_\infty)\end{align*} $$

is the inclusion of components corresponding to morphisms $\mathbf {D}_1\times ... \times \mathbf {D}_n\to \mathbf {D}_\infty $ lying over a morphism $\mathbf {C}_1\times ... \times \mathbf {C}_n\to \mathbf {C}_\infty $ and sending (factor-wise) coCartesian edges to coCartesian edges.

The two pullback squares above and the commutative triangle induce the following pullback square:

The description of $\mathbf {coCart}^{\times } \subset \mathbf {Cat} ^{\times }$ above clearly specializes to the claimed description of $\mathbf {coCart}_{\mathbf {C}}^{\otimes }\subset (\mathbf {Cat} _{/\mathbf {C}})^{\otimes }$ .

Proof. $\mathbf {coCart}_{\mathbf {C}} \subset \mathbf {Cat} _{/\mathbf {C}}$ is a non-full sub- ${\mathcal {O}}$ -operad, so in particular, $\mathrm {Alg}_{\mathcal {O}}(\mathbf {coCart}_{\mathbf {C}})\to \mathrm {Alg}_{\mathcal {O}}(\mathbf {Cat} _{/\mathbf {C}})$ is a non-full subcategory.

By [Reference Antolìn-Camarena and Barthel1, Lemma 2.12], $\mathrm {Alg}_{\mathcal {O}}(\mathbf {Cat} _{/\mathbf {C}}) \simeq \mathrm {Alg}_{\mathcal {O}}(\mathbf {Cat} )_{/\mathbf {C}}$ and by the equivalence between $\mathrm {Alg}_{\mathcal {O}}(\mathbf {Cat} )$ and ${\mathcal {O}}$ -monoidal categories, the latter is a non-full subcategory of $\mathbf {Cat} _{/\mathbf {C}^{\otimes }}$ .

This chain of functors witnesses $\mathrm {Alg}_{\mathcal {O}}(\mathbf {coCart}_{\mathbf {C}})$ as a non-full subcategory of $\mathbf {Cat} _{/\mathbf {C}^{\otimes }}$ . Examining each stage, the essential image yields the desired claim.

Proof. This follows from Proposition 5.5 and the description of ${\mathcal {O}}$ -algebras in the Day convolution monoidal structure.

Warning 6.1. In this section, some of the statements will not be completely proved, only conditional on some other statements. The conditional statements which we state nonetheless will be appended with a (*) symbol.

Observation 6.2. Let $f:\mathbf {C}\to \mathbf {Cat} $ be as above. The microcosmic construction makes $\mathbf {D}^{\otimes }$ fit in a pullback

In particular, because $\mathbf {D}^{\otimes }$ and the abstract construction from the macrocosmic version agree at the level of underlying categories, it would suffice to construct a similar commutative square with the latter as the top left-hand corner, for then we would get a comparison map which would be an equivalence of underlying categories, and hence an equivalence.

The issue in doing this with our current approach is that we want to use the naturality of our equivalence, but we only get naturality in strong symmetric monoidal functors, because we only have control over commutative algebras in $\mathbf {Cat} $ —but $\mathbf {C}\to \mathbf {Cat} $ is only lax symmetric monoidal.

Proof of Corollary 6.3

Observe that $\mathrm {CAlg}(\widehat {\mathbf {coCart}}) \to \mathrm {CAlg}(\operatorname {Fun}(\Delta ^1,\widehat {\mathbf {Cat} }))$ is a morphism of Cartesian fibrations over $\mathrm {CAlg}(\widehat {\mathbf {Cat} })$ [Reference Horel and Ramzi13, Theorem B.1], and the former is a non-full subcategory of the latter.

In particular, taking $f:\mathbf {C}\to \mathbf {Cat} $ as a morphism in the base, and taking a Cartesian lift in $\mathrm {CAlg}(\widehat {\mathbf {coCart}})$ with target , we might as well take this Cartesian lift in $\mathrm {CAlg}(\operatorname {Fun}(\Delta ^1,\widehat {\mathbf {Cat} })) \simeq \operatorname {Fun}(\Delta ^1,\mathrm {CAlg}(\widehat {\mathbf {Cat} }))$ .

This Cartesian lift is then a morphism in this arrow category, hence a commutative square

of symmetric monoidal categories and (strong !) symmetric monoidal functors.

In particular, by Observation 6.2, it really suffices to show that this $\mathbf {D}$ that we obtain is the same as the one from using the construction following the macrocosm version, Theorem B.

For this, we observe that by Theorem A, we have a commutative square of symmetric monoidal categories of the form

where the horizontal arrows are given by the transformation from Theorem A.

Because the horizontal functors are equivalences, we can take right adjoints of the vertical maps and still have a commutative diagram, now of lax symmetric monoidal functors:

Following along from $\mathbf {C}\mapsto \mathbf {C}$ in the bottom left-hand corner, up-right gives f and then the symmetric monoidal structure on $\mathbf {D}$ from construction 2. above.

Going right-up gives with its unique Cartesian symmetric monoidal structure, and then the pullback of this. We are reduced to proving that this pullback coincides with the pullbacks from Cartesian liftings from the beginning of the proof. By Lemma 4.14, this follows from the next (conditional) lemma.

Proof of Lemma 6.5

Here is a reformulation of the statement: $\mathbf {E}\to \mathbf {F}$ induces a lax symmetric monoidal functor $\mathbf {F}\to \mathbf {Cat} $ , so we get a functor $\mathrm {CAlg}(\mathbf {F})\to \mathrm {CAlg}(\mathbf {Cat} )$ . We can further postcompose with $\mathrm {CAlg}(\mathbf {Cat} )\overset {\mathrm {CAlg}(-)}\to \mathbf {Cat} $ and observe that our hypotheses actually make this factor through $\mathrm {Adj}^L$ , the category of categories and left adjoint functors.

One then uses the equivalence $\mathrm {Adj}^L\simeq (\mathrm {Adj}^R)^{\mathrm {op}}$ and forgets back down to $\mathbf {Cat} $ to get a functor $\mathrm {CAlg}(\mathbf {F})^{\mathrm {op}}\to \mathbf {Cat} $ , which is classified by a Cartesian fibration $\mathbf {P}\to \mathrm {CAlg}(\mathbf {F})$ .

To prove the lemma, it is sufficient to prove that $\mathbf {P}\simeq \mathrm {CAlg}(\mathbf {E})$ as Cartesian fibrations over $\mathrm {CAlg}(\mathbf {F})$ .

Because both are also coCartesian fibrations, it suffices to prove it for the corresponding coCartesian fibration. For $\mathrm {CAlg}(\mathbf {E})$ , we do not have anything to change, but for $\mathbf {P}$ , we can simplify the above construction and not use the $\mathrm {Adj}^L\simeq (\mathrm {Adj}^R)^{\mathrm {op}}$ equivalence.

In other words, it suffices to prove that we have a pullback square

because there is one for $\mathbf {P}$ .

But by Lemma 4.14, the fiber is the correct one, so in fact it suffices to find a suitable commutative square.

Because we have a commutative square of lax symmetric monoidal functors of the form:

by applying $\mathrm {CAlg}(-)$ to it, we see that it suffices to exhibit a suitable commutative square of the form

that is, the previous kind of diagram but in the universal case (note that the bottom map is not the forgetful functor, but the functor $\mathbf {C}\mapsto \mathrm {CAlg}(\mathbf {C})$ ).

But this is exactly Lemma 6.4, that is, the lemma that we did not prove. This is why this statement is conditional.

Warning 6.7. The $^{co}$ appearing up there adds an extra layer of difficulty. It stems from the fact that $\operatorname {Fun}(\mathbf {C},\mathbf {D})^{\mathrm {op}}\to \operatorname {Fun}(\operatorname {Fun}(\mathbf {C},\mathbf {Cat} ),\operatorname {Fun}(\mathbf {D},\mathbf {Cat} )), f\mapsto f_!$ is contravariant (because $\operatorname {Fun}(\mathbf {C}^{\mathrm {op}},\mathbf {D}^{\mathrm {op}})\simeq \operatorname {Fun}(\mathbf {C},\mathbf {D})^{\mathrm {op}}$ , not $\operatorname {Fun}(\mathbf {C},\mathbf {D})$ ). In particular, a lax symmetric monoidal functor $\mathbf {C}\to \mathbf {D}$ induces an oplax symmetric monoidal functor $\operatorname {Fun}(\mathbf {C},\mathbf {Cat} )\to ~\operatorname {Fun}(\mathbf {D},\mathbf {Cat} )$ , while restriction is lax symmetric monoidal.

One might think that this difficulty is artificial, as it goes away when looking at Cartesian fibrations and contravariant functor $\mathbf {C}^{\mathrm {op}}\to \mathbf {Cat} $ , but the issue with this then becomes that a lax symmetric monoidal functor $\mathbf {C}\to \mathbf {D}$ is oplax symmetric monoidal as a functor $\mathbf {C}^{\mathrm {op}}\to \mathbf {D}^{\mathrm {op}}$ , so this difficulty is here regardless.