Notation 3.3. Let

$$ \begin{align*}{\mathcal R} \subset \mathrm{Fun}([1],\infty\mathrm{Cat})\end{align*} $$

be the full subcategory of right fibrations and

$$ \begin{align*}{\mathcal U} \subset {\mathcal R}\end{align*} $$

the full subcategory of representable right fibrations.

Proof. The functor $\gamma : \mathrm {Fun}([1],\infty \mathrm {Cat}) \to \infty \mathrm {Cat}$ evaluating at the target is a cartesian fibration because $\infty \mathrm {Cat}$ admits pullbacks. Since right fibrations are stable under pullback, $\gamma $ restricts to a cartesian fibration $\rho : {\mathcal R} \to \infty \mathrm {Cat}$ with the same cartesian morphisms. For any functor $\psi : {\mathcal C} \to {\mathcal D}$ the fiber transport ${\mathcal R}_{\mathcal D} \to {\mathcal R}_{\mathcal C}$ of ${\mathcal R} \to \infty \mathrm {Cat}$ admits a left adjoint $\psi _!$ that takes the left Kan extension along the functor ${\mathcal C}^{\mathrm {op}} \to {\mathcal D}^{\mathrm {op}}.$ This guarantees that the cartesian fibration $\rho : {\mathcal R} \to \infty \mathrm {Cat}$ is also a cocartesian fibration. Since $\psi _!$ preserves representables, the cocartesian fibration $\rho : {\mathcal R} \to \infty \mathrm {Cat}$ restricts to a cocartesian fibration $\kappa : {\mathcal U} \to \infty \mathrm {Cat}$ with the same cocartesian morphisms.

Notation 3.6. Let ${\mathcal P}: \infty \mathrm {Cat} \to \infty \widehat {\mathrm {Cat}}$ be the functor classifying the cocartesian fibration $\rho : {\mathcal R} \to \infty \mathrm {Cat}$ .

Notation 3.7. For every functor ${\mathcal A} \to {\mathcal B}$ and $\infty $ -category ${\mathrm K}$ let ${\mathcal A}^{\mathrm K} \to {\mathcal B}$ be the pullback ${\mathcal B} \times _{\mathrm {Fun}({\mathrm K},{\mathcal B})} \mathrm {Fun}({\mathrm K},{\mathcal A}) $ along the diagonal functor ${\mathcal B} \to \mathrm {Fun}({\mathrm K},{\mathcal B}).$

Proof. (1): Let ${\mathcal V} \to \infty \mathrm {Cat}$ be the cocartesian fibration classifying the identity. To prove (1) we will construct an equivalence ${\mathcal V} \simeq {\mathcal U}$ over $\infty \mathrm {Cat}$ . By the Yoneda-lemma it will suffice to show that for any $\infty $ -category ${\mathcal A}$ and functor $\theta : {\mathcal A} \to \infty \mathrm {Cat}$ there is a bijection between the sets of equivalence classes of functors ${\mathcal A} \to {\mathcal V}$ over $\infty \mathrm {Cat}$ and ${\mathcal A} \to {\mathcal U}$ over $\infty \mathrm {Cat}$ .

Let ${\mathcal C} \to {\mathcal A}$ be the cocartesian fibration classified by the functor $\theta : {\mathcal A} \to \infty \mathrm {Cat}$ . By naturality of the Grothendieck construction there is a canonical equivalence ${\mathcal C} \simeq {\mathcal A} \times _{\infty \mathrm {Cat}}{\mathcal V}$ over ${\mathrm A}.$ Thus a functor ${\mathcal A} \to {\mathcal V}$ over $\infty \mathrm {Cat}$ corresponds to a functor ${\mathcal A} \to {\mathcal A} \times _{\infty \mathrm {Cat}}{\mathcal V} \simeq {\mathcal C}$ over ${\mathrm A}, $ that is, a section of the cocartesian fibration ${\mathcal C} \to {\mathcal A}$ .

Next we identify functors ${\mathcal A} \to {\mathcal U} $ over $\infty \mathrm {Cat}$ . A functor ${\mathcal A} \to \mathrm {Fun}([1],\infty \mathrm {Cat})$ over $\infty \mathrm {Cat}$ is classified by a map of cocartesian fibrations ${\mathcal D} \to {\mathcal C}$ over ${\mathcal A}$ . Since ${\mathcal U} $ is the full subcategory of $\mathrm {Fun}([1],\infty \mathrm {Cat})$ spanned by the representable right fibrations, a functor ${\mathcal A} \to {\mathcal U} \subset \mathrm {Fun}([1],\infty \mathrm {Cat})$ over $\infty \mathrm {Cat}$ is classified by a map of cocartesian fibrations ${\mathcal D} \to {\mathcal C}$ over ${\mathcal A}$ that induces on the fiber over any ${\mathrm A} \in {\mathcal A}$ a representable right fibration.

Thus, to complete the proof, we will show that there is a bijection between equivalence classes of sections of the cocartesian fibration ${\mathcal C} \to {\mathcal A}$ and maps of cocartesian fibrations ${\mathcal D} \to {\mathcal C}$ over ${\mathcal A}$ that induce on the fiber over any ${\mathrm A} \in {\mathcal A}$ a representable right fibration, where ${\mathcal D} \to {\mathcal A}$ is some cocartesian fibration.

A section $\alpha $ of ${\mathcal C} \to {\mathcal A}$ gives rise to a map ${\mathcal A} \times _{{\mathcal C}^{\{1\}} } {\mathcal C}^{[1]} \to {\mathcal C}^{\{0\}}$ of cocartesian fibrations over ${\mathcal A}$ that induces on the fiber over any ${\mathrm A} \in {\mathcal A}$ the representable right fibration $({\mathcal C}_{\mathrm A})_{/\alpha ({\mathrm A})} \to {\mathcal C}_{\mathrm A}$ .

Conversely, let $\psi : {\mathcal D} \to {\mathcal C}$ be a map of cocartesian fibrations over ${\mathcal A}$ such that for any ${\mathrm A} \in {\mathcal A}$ the fiber ${\mathcal D}_{\mathrm A}$ has a final object. Then the $\infty $ -category $\mathrm {Fun}_{\mathcal A}({\mathcal A},{\mathcal D})$ has a final object $\beta $ such that for any ${\mathrm A} \in {\mathcal A}$ the image $\beta ({\mathrm A})$ is final in ${\mathcal D}_{\mathrm A}.$ The composition $\psi \circ \beta : {\mathcal A} \to {\mathcal C}$ is a section of ${\mathcal C} \to {\mathcal A}$ .

We will prove that sending a section $\alpha $ of ${\mathcal C} \to {\mathcal A}$ to the map ${\mathcal A} \times _{{\mathcal C}^{\{1\}} } {\mathcal C}^{[1]} \to {\mathcal C}^{\{0\}}$ of cocartesian fibrations over ${\mathcal A}$ is inverse to sending a map $\psi : {\mathcal D} \to {\mathcal C} $ of cocartesian fibrations over ${\mathcal A}$ such that for any ${\mathrm A} \in {\mathcal A}$ the fiber ${\mathcal D}_{\mathrm A}$ has a final object, to the section $\psi \circ \beta : {\mathcal A} \to {\mathcal C}$ of ${\mathcal C} \to {\mathcal A}$ .

The map ${\mathcal A} \times _{{\mathcal C}^{\{1\}} } {\mathcal C}^{[1]} \to {\mathcal C}^{\{0\}}$ of cocartesian fibrations over ${\mathcal A}$ induces on sections the functor $\mathrm {Fun}_{\mathcal A}({\mathcal A},{\mathcal C})_{/\alpha } \to \mathrm {Fun}_{\mathcal A}({\mathcal A},{\mathcal C})$ . Consequently, $\alpha : {\mathcal A} \to {\mathcal C} $ is the associated section of the map ${\mathcal A} \times _{{\mathcal C}^{\{1\}} } {\mathcal C}^{[1]} \to {\mathcal C}^{\{0\}}$ of cocartesian fibrations over ${\mathcal A}$ .

It remains to see that for any map ${\mathcal D} \to {\mathcal C}$ of cocartesian fibrations over ${\mathcal A}$ that induces on the fiber over any ${\mathrm A} \in {\mathcal A}$ a representable right fibration, with associated section $\beta : {\mathcal A} \to {\mathcal C},$ there is an equivalence ${\mathcal D} \simeq {\mathcal A} \times _{{\mathcal C}^{\{1\}} } {\mathcal C}^{[1]} \to {\mathcal C}^{\{0\}} $ over ${\mathcal C}.$

Let $\psi : {\mathcal D} \to {\mathcal C}$ be a map of cocartesian fibrations over ${\mathcal A}$ that induces on the fiber over any ${\mathrm A} \in {\mathcal A}$ a representable right fibration. The induced map

(3.5) $$ \begin{align} {\mathcal A} \times_{{\mathcal D}^{\{1\}} } {\mathcal D}^{[1]} \to {\mathcal D}^{\{0\}} \end{align} $$

of cocartesian fibrations over ${\mathcal A}$ induces on the fiber over any ${\mathrm A} \in {\mathcal A}$ the functor $({\mathcal D}_{\mathrm A})_{/\beta ({\mathrm A})} \to {\mathcal D}_{\mathrm A}$ , which is an equivalence since $\beta ({\mathrm A})$ is a final object in ${\mathcal D}_{\mathrm A}.$ Therefore the functor (3.5) is an equivalence.

As we have noted, the $\infty $ -category $\mathrm {Fun}_{\mathcal A}({\mathcal A},{\mathcal D})$ admits a final object $\beta $ such that for any ${\mathrm A} \in {\mathcal A}$ the image $\beta ({\mathrm A})$ is final in ${\mathcal D}_{\mathrm A}.$ The section $\psi \circ \beta : {\mathcal A} \to {\mathcal C}$ of ${\mathcal C} \to {\mathcal A}$ and the map $\psi : {\mathcal D} \to {\mathcal C}$ of cocartesian fibrations over ${\mathcal A}$ give rise to a map

(3.6) $$ \begin{align} {\mathcal A} \times_{{\mathcal D}^{\{1\}} } {\mathcal D}^{[1]} \to {\mathcal A} \times_{{\mathcal C}^{\{1\}} } {\mathcal C}^{[1]} \end{align} $$

of cocartesian fibrations over ${\mathcal A}$ , which induces on the fiber over any ${\mathrm A} \in {\mathcal A}$ the functor $({\mathcal D}_{\mathrm A})_{/\beta ({\mathrm A})} \to ({\mathcal C}_{\mathrm A})_{/\alpha ({\mathrm A})}$ . Since $\psi : {\mathcal D} \to {\mathcal C}$ induces on the fiber over A a right fibration ${\mathcal D}_{\mathrm A} \to {\mathcal C}_{\mathrm A}$ , the functor $({\mathcal D}_{\mathrm A})_{/\beta ({\mathrm A})} \to ({\mathcal C}_{\mathrm A})_{/\alpha ({\mathrm A})}$ is an equivalence. Consequently, the functor (3.6) is an equivalence.

Via the equivalences (3.5) and (3.6) the functor ${\mathcal D} \to {\mathcal C}$ is equivalent over ${\mathcal C}$ to ${\mathcal A} \times _{{\mathcal C}^{\{1\}} } {\mathcal C}^{[1]} \to {\mathcal C}^{\{0\}}$ . The proof of (1) shows (2).

Proof. By [Reference Heine17, Lemma A.7.] the embedding $\theta : {\mathcal C} \subset {\mathrm {Env}}({\mathcal C})$ admits a left adjoint ${\mathrm L}$ relative to ${\mathcal A}$ , which is a map of cocartesian fibrations over ${\mathcal A}$ . The relative left adjoint gives rise to a map ${\mathrm L}_!: {\mathcal P}^{\mathcal A}({\mathrm {Env}}({\mathcal C})) \to {\mathcal P}^{\mathcal A}({\mathcal C})$ of cocartesian fibrations over ${\mathcal A}$ . For every ${\mathrm A} \in {\mathcal A}$ the induced localization ${\mathrm L}_{\mathrm A}: {\mathrm {Env}}({\mathcal C})_{\mathrm A} \rightleftarrows {\mathcal C}_{\mathrm A}: \theta _{\mathrm A}$ induces a localization $({\mathrm L}_!)_{\mathrm A} \simeq ({\mathrm L}_{\mathrm A})_! : {\mathcal P}({\mathrm {Env}}({\mathcal C})_{\mathrm A})\rightleftarrows {\mathcal P}({\mathcal C}_{\mathrm A}): (\theta _{\mathrm A})_! $ . By [Reference Lurie26, Proposition 7.3.2.6.] this implies that the map $ {\mathrm L}_!: {\mathcal P}^{\mathcal A}({\mathrm {Env}}({\mathcal C})) \to {\mathcal P}^{\mathcal A}({\mathcal C})$ of cocartesian fibrations over ${\mathcal A}$ admits a fully faithful right adjoint relative to ${\mathcal A}$ that induces on the fiber over ${\mathrm A} \in {\mathcal A}$ the embedding $(\theta _{\mathrm A})_!.$

Proof. For every ${\mathrm Z} \in {\mathcal C}$ lying over ${\mathrm B} \in {\mathcal A}$ the canonical map

$$ \begin{align*}{\mathcal C}_{\mathrm B}({\mathrm Y}',{\mathrm Z}) \simeq {\mathcal D}_{\mathrm B}({\mathrm Y},{\mathrm Z}) \to \{{\mathrm{f}}\}\times_{{\mathcal A}({\mathrm A}, {\mathrm B})} {\mathcal D}({\mathrm X},{\mathrm Z})\end{align*} $$

is an equivalence.

Proof. The functor ${\mathcal P}^{\mathcal A}({\mathrm {Env}}({\mathcal C})) \to {\mathcal A} $ is a cartesian fibration and for every ${\mathrm A} \in {\mathcal A}$ the induced embedding $(\theta _{\mathrm A})_!: {\mathcal P}^{\mathcal A}({\mathcal C})_{\mathrm A} \simeq {\mathcal P}({\mathcal C}_{\mathrm A}) \subset {\mathcal P}^{\mathcal A}({\mathrm {Env}}({\mathcal C}))_{\mathrm A} \simeq {\mathcal P}({\mathrm {Env}}({\mathcal C})_{\mathrm A})$ admits a right adjoint. Thus by Lemma 3.13 the restriction ${\mathcal P}^{\mathcal A}({\mathcal C}) \subset {\mathcal P}^{\mathcal A}({\mathrm {Env}}({\mathcal C})) \to {\mathcal A}$ is a locally cartesian fibration.

Notation 3.15. Let ${\mathcal C} \to {\mathcal A}, {\mathcal D} \to {\mathcal B}$ be functors. Let

$$ \begin{align*}\Theta({\mathcal C},{\mathcal D}) := \mathrm{Fun}({\mathcal A},{\mathcal B}) \times_{\mathrm{Fun}({\mathcal C},{\mathcal B})} \mathrm{Fun}({\mathcal C}, {\mathcal D})\end{align*} $$

and

$$ \begin{align*}\Theta({\mathcal C},{\mathcal D})^{\mathrm L} \subset \Theta({\mathcal C},{\mathcal D})\end{align*} $$

the full subcategory spanned by the commutative squares

such that for every ${\mathrm A} \in {\mathcal A}$ the induced functor ${\mathcal C}_{\mathrm A} \to {\mathcal D}_{\psi ({\mathrm A})}$ on the fiber over ${\mathrm A}$ preserves small colimits.

Proof. (2) is similar to (1). We prove (1). We first reduce to the case that ${\mathcal D} \to {\mathcal B}$ is a cocartesian and cartesian fibration whose fibers admit large colimits.

By the opposite of Lemma 3.17 (applied to a larger universe) there is an embedding ${\mathcal D} \subset {\mathcal E}$ over ${\mathcal B}$ that induces on the fiber over every object of ${\mathcal B}$ a small colimits preserving functor, where ${\mathcal E} \to {\mathcal B}$ is a cocartesian and cartesian fibration whose fibers admit large colimits and are locally large. Since the $\infty $ -category of presheaves is generated by the representable presheaves under small colimits [Reference Heine15, Lemma 2.10.], the functor $\theta _{\mathcal D}$ is the pullback of the functor $\theta _{\mathcal E}$ . So it is enough to check that $\theta _{\mathcal E}$ is an equivalence and we can assume that ${\mathcal D} \to {\mathcal B}$ is a cocartesian and cartesian fibration whose fibers admit large colimits and are locally large.

Next we further reduce the statement to show that for every cocartesian and cartesian fibration ${\mathcal D} \to {\mathcal A}$ whose fibers admit large colimits and are locally large, the canonical functor

(3.7) $$ \begin{align} \kappa_{\mathcal D}: \mathrm{Fun}_{\mathcal A}^{\mathrm L}({\mathcal P}^{\mathcal A}({\mathcal C}), {\mathcal D}) \to \mathrm{Fun}_{\mathcal A}({\mathcal C}, {\mathcal D})\end{align} $$

is an equivalence, where the left hand side is the full subcategory of functors over ${\mathcal A}$ that induce on every fiber a small colimits preserving functor.

There is a commutative triangle

Because the functor ${\mathcal D} \to {\mathcal B}$ is a cocartesian and cartesian fibration, this triangle is a map of cocartesian and cartesian fibrations over $\mathrm {Fun}({\mathcal A},{\mathcal B}).$ Since the functor ${\mathcal D} \to {\mathcal B}$ is a cocartesian and cartesian fibration, the fiber transports of the cocartesian fibration are left adjoints. This guarantees that the restriction $\Theta ({\mathcal P}^{\mathcal A}({\mathcal C}), {\mathcal D})^{\mathrm L} \to \mathrm {Fun}({\mathcal A},{\mathcal B})$ is also a cocartesian fibration and the embedding $\Theta ({\mathcal P}^{\mathcal A}({\mathcal C}), {\mathcal D})^{\mathrm L} \subset \Theta ({\mathcal P}^{\mathcal A}({\mathcal C}), {\mathcal D})$ is a map of cocartesian fibrations over $\mathrm {Fun}({\mathcal A},{\mathcal B}) $ . Hence also the restriction

is a map of cocartesian fibrations over $\mathrm {Fun}({\mathcal A},{\mathcal B}) $ .

Consequently, $\theta _{\mathcal D}$ is an equivalence if it induces on the fiber over every functor $\psi : {\mathcal A} \to {\mathcal B}$ an equivalence. The functor $\theta _{\mathcal D}$ induces on the fiber over $\psi $ the functor

$$ \begin{align*}\kappa_{{\mathcal A} \times_{\mathcal B} {\mathcal D}}: \mathrm{Fun}_{\mathcal A}^{\mathrm L}({\mathcal P}^{\mathcal A}({\mathcal C}), {\mathcal A} \times_{\mathcal B} {\mathcal D}) \to \mathrm{Fun}_{\mathcal A}({\mathcal C}, {\mathcal A} \times_{\mathcal B} {\mathcal D}).\end{align*} $$

Therefore it is enough to see that the functor (3.7) is an equivalence.

The functor (3.7) is conservative since for every ${\mathrm A} \in {\mathcal A}$ the $\infty $ -category ${\mathcal P}^{\mathcal A}({\mathcal C})_{\mathrm A} \simeq {\mathcal P}({\mathcal C}_{\mathrm A})$ is generated by ${\mathcal C}_{\mathrm A}$ under small colimits. Therefore, the functor (3.7) is an equivalence if it admits a fully faithful left adjoint. We devote the remaining proof to showing this.

By Proposition 3.14 the functor ${\mathcal P}^{\mathcal A}({\mathcal C}) \to {\mathcal A}$ is a locally cartesian fibration. This implies that ${\mathcal P}^{\mathcal A}({\mathcal C})$ is large because ${\mathcal A}$ is large and the fibers of ${\mathcal P}^{\mathcal A}({\mathcal C}) \to {\mathcal A}$ are large. Let $\iota : {\mathcal C} \subset {\mathcal P}^{\mathcal A}({\mathcal C})$ be the canonical embedding. Since ${\mathcal P}^{\mathcal A}({\mathcal C})$ is large and the fibers of the cocartesian fibration ${\mathcal D} \to {\mathcal A}$ admit large colimits preserved by the fiber transports, by [Reference Lurie27, Corollary 4.3.2.14., Proposition 4.3.2.17.] the induced functor

$$ \begin{align*}\iota^*: \mathrm{Fun}_{\mathcal A}({\mathcal P}^{\mathcal A}({\mathcal C}), {\mathcal D}) \to \mathrm{Fun}_{\mathcal A}({\mathcal C}, {\mathcal D})\end{align*} $$

admits a fully faithful left adjoint $\iota _!$ , which forms the left Kan extension relative to ${\mathcal A}$ along the embedding ${\mathcal C} \to {\mathcal P}^{\mathcal A}({\mathcal C}).$ We complete the proof by showing that the fully faithful left adjoint $\iota _!$ lands in the full subcategory $\mathrm {Fun}_{\mathcal A}({\mathcal P}^{\mathcal A}({\mathcal C}), {\mathcal D})^{\mathrm L},$ and so is a fully faithful left adjoint of $\kappa _{\mathcal D}$ . In other words, we have to see that the left Kan extension $ {\mathcal P}^{\mathcal A}({\mathcal C}) \to {\mathcal D}$ relative to ${\mathcal A}$ of any functor ${\mathcal C} \to {\mathcal D}$ over ${\mathcal A}$ preserves fiberwise small colimits.

Let ${j}: {\mathcal P}^{\mathcal A}({\mathcal C}) \subset {\mathcal P}^{\mathcal A}({\mathrm {Env}}({\mathcal C}))$ be the canonical embedding. Because ${\mathcal P}^{\mathcal A}({\mathrm {Env}}({\mathcal C}))$ is large, by [Reference Lurie27, Corollary 4.3.2.14., Proposition 4.3.2.17.] the induced functor

$$ \begin{align*}{j}^*: \mathrm{Fun}_{\mathcal A}({\mathcal P}^{\mathcal A}({\mathrm{Env}}({\mathcal C})), {\mathcal D}) \to \mathrm{Fun}_{\mathcal A}({\mathcal P}^{\mathcal A}({\mathcal C}), {\mathcal D})\end{align*} $$

admits a fully faithful left adjoint ${j}_!, $ which forms the left Kan extension relative to ${\mathcal A}$ along the embedding ${\mathcal P}^{\mathcal A}({\mathcal C}) \to {\mathcal P}^{\mathcal A}({\mathrm {Env}}({\mathcal C})).$ Since ${j}_!$ is fully faithful, the unit $\mathrm {id} \to {j}^*{j}$ is an equivalence.

We want to see that for every functor ${\mathcal C} \to {\mathcal D}$ over ${\mathcal A}$ the left Kan extension $\iota _!({\mathrm F}) \simeq {j}^*({j}_!(\iota _!({\mathrm F}))): {\mathcal P}^{\mathcal A}({\mathcal C}) \to {\mathcal D}$ relative to ${\mathcal A}$ preserves fiberwise small colimits. The embedding ${j}: {\mathcal P}^{\mathcal A}({\mathcal C}) \subset {\mathcal P}^{\mathcal A}({\mathrm {Env}}({\mathcal C}))$ preserves fiberwise small colimits. Hence it is enough to see that ${j}_!(\iota _!({\mathrm F}))$ preserves fiberwise small colimits.

The functor ${j} \circ \iota : {\mathcal C} \to {\mathcal P}^{\mathcal A}({\mathcal C}) \subset {\mathcal P}^{\mathcal A}({\mathrm {Env}}({\mathcal C})) $ factors as ${\mathcal C} \to {\mathrm {Env}}({\mathcal C}) \subset {\mathcal P}^{\mathcal A}({\mathrm {Env}}({\mathcal C})). $ Therefore by adjointness, the functor ${j}_!(\iota _!({\mathrm F}))$ is likewise the left Kan extension of ${\mathrm F}'$ relative to ${\mathcal A}$ along the embedding ${i}:{\mathrm {Env}}({\mathcal C}) \subset {\mathcal P}^{\mathcal A}({\mathrm {Env}}({\mathcal C}))$ , where ${\mathrm F}'$ is the left Kan extension of ${\mathrm F}: {\mathcal C} \to {\mathcal A} $ relative to ${\mathcal A}$ along the embedding ${\mathcal C} \to {\mathrm {Env}}({\mathcal C})$ .

So it suffices to verify that the left Kan extension ${i}_!({\mathrm H}): {\mathcal P}^{\mathcal A}({\mathrm {Env}}({\mathcal C})) \to {\mathcal D} $ relative to ${\mathcal A}$ of any functor ${\mathrm H}: {\mathrm {Env}}({\mathcal C}) \to {\mathcal D}$ over ${\mathcal A}$ preserves fiberwise small colimits.

Because ${i}: {\mathrm {Env}}({\mathcal C}) \subset {\mathcal P}^{\mathcal A}({\mathrm {Env}}({\mathcal C}))$ is a map of cocartesian fibrations over ${\mathcal A}$ , by [Reference Lurie27, Proposition 4.3.3.10.] for every ${\mathrm A} \in {\mathcal A}$ the induced functor

$$ \begin{align*}{i}_!({\mathrm H})_{\mathrm A}: {\mathcal P}^{\mathcal A}({\mathrm{Env}}({\mathcal C}))_{\mathrm A} \simeq {\mathcal P}({\mathrm{Env}}({\mathcal C})_{\mathrm A}) \to {\mathcal D}_{\mathrm A}\end{align*} $$

on the fiber over ${\mathrm A}$ is computed as the left Kan extension of ${\mathrm H}_{\mathrm A}: {\mathrm {Env}}({\mathcal C})_{\mathrm A} \to {\mathcal D}_{\mathrm A}$ along the induced embedding $ {\mathrm {Env}}({\mathcal C})_{\mathrm A} \subset {\mathcal P}({\mathrm {Env}}({\mathcal C})_{\mathrm A})$ , which by [Reference Lurie27, Lemma 5.1.5.5.] preserves small colimits.

Proof. Since ${\mathcal D} \to {\mathcal A}$ is a locally cocartesian fibration, by [Reference Heine17, Lemma A.7.] the embedding ${\mathcal D} \subset {\mathrm {Env}}({\mathcal D})$ induces on the fiber over every object of ${\mathcal A}$ a right adjoint functor. Since ${\mathcal D} \to {\mathcal A}$ is a locally cocartesian fibration of small $\infty $ -categories, also ${\mathcal D}$ is small. Take ${\mathcal E}:= {\mathcal P}^{\mathcal A}({\mathrm {Env}}({\mathcal D})).$ The embedding ${\mathcal D} \subset {\mathrm {Env}}({\mathcal D}) \to {\mathcal P}^{\mathcal A}({\mathrm {Env}}({\mathcal D}))$ induces on every fiber a small limits preserving functor.

Proof. We first prove that the embedding ${\mathcal C} \subset {\mathcal P}^{\mathcal A}({\mathcal C}) $ is a map of locally cartesian fibrations over ${\mathcal A}$ if ${\mathcal C} \to {\mathcal A}$ is a locally cartesian fibration. By Remark 3.10 for every functor ${\mathcal A}' \to {\mathcal A}$ there is a canonical equivalence

$$ \begin{align*}{\mathcal A}' \times_{ {\mathcal A}} {\mathcal P}^{{\mathcal A}}({\mathcal C}) \simeq {\mathcal P}^{{\mathcal A}'}( {\mathcal A}' \times_{{\mathcal A}} {\mathcal C}) \end{align*} $$

over ${\mathcal A}'$ so that we can reduce to the case ${\mathcal A}=[1].$ Let ${\mathcal M} \to [1]$ be a functor and ${\mathcal C}:={\mathcal M}_0,{\mathcal D}:={\mathcal M}_1$ the fibers. The functor ${\mathcal P}({\mathcal M})^{[1]} \to [1]$ is a cartesian fibration classifying the functor $\phi : {\mathcal P}({\mathcal D}) \subset {\mathcal P}({\mathcal M}) \to {\mathcal P}({\mathcal C})$ , where the first functor takes left Kan extension along the embedding ${\mathcal D} \subset {\mathcal M}$ and the second functor restricts along the embedding ${\mathcal C} \subset {\mathcal M}.$ If ${\mathcal M}\to [1]$ is a cartesian fibration classifying a functor ${\mathrm G}: {\mathcal D} \to {\mathcal C},$ then $\phi $ is the extension of the functor ${\mathcal D} \xrightarrow {{\mathrm G}} {\mathcal C} \subset {\mathcal P}({\mathcal C}).$ So the embedding ${\mathcal M} \subset {\mathcal P}({\mathcal M})^{[1]}$ is a map of cartesian fibrations over $[1].$

A locally cartesian fibration $\phi $ is a cartesian fibration if and only if the collection of $\phi $ -cartesian morphisms is closed under composition. Hence ${\mathcal C} \to {\mathcal A}$ is a cartesian fibration if the functor $ {\mathcal P}^{\mathcal A}({\mathcal C}) \to {\mathcal A}$ is a cartesian fibration. Conversely, if ${\mathcal C} \to {\mathcal A}$ is a cartesian fibration, the functor $ {\mathcal P}^{\mathcal A}({\mathcal C}) \to {\mathcal A}$ is a cartesian fibration since the fibers of ${\mathcal P}^{\mathcal A}({\mathcal C}) \to {\mathcal A}$ are generated by the representable presheaves under small colimits, and the fiber transports of ${\mathcal P}^{\mathcal A}({\mathcal C}) \to {\mathcal A}$ preserve small colimits.

Proof. By Lemma 3.18 for any locally cartesian fibration ${\mathcal C} \to {\mathcal A}$ the functor ${\mathcal P}^{\mathcal A}({\mathcal C}) \to {\mathcal A}$ is a locally cartesian fibration and the embedding ${\mathcal C} \to {\mathcal P}^{\mathcal A}({\mathcal C}) $ is a map of locally cartesian fibrations over ${\mathcal A}$ . By Lemma 3.11 and Remark 3.10 for every locally cocartesian fibration ${\mathcal C} \to {\mathcal A}$ the functor ${\mathcal P}^{\mathcal A}({\mathcal C}) \to {\mathcal A}$ is a locally cocartesian and locally cartesian fibration and the embedding ${\mathcal C} \to {\mathcal P}^{\mathcal A}({\mathcal C}) $ is a map of locally cocartesian fibrations over ${\mathcal A}$ . Since the fibers of ${\mathcal P}^{\mathcal A}({\mathcal C}) \to {\mathcal A}$ are generated under small colimits by the representables and the fiber transports of the locally (co)cartesian fibration ${\mathcal P}^{\mathcal A}({\mathcal C}) \to {\mathcal A}$ preserve small colimits by Proposition 3.14, the induced functor ${\mathcal P}^{\mathcal A}({\mathcal C}) \to {\mathcal P}^{\mathcal A}({\mathcal D})$ is a map of locally (co)cartesian fibrations over ${\mathcal A}$ .

Notation 3.21. For every exponentiable fibration $\phi : {\mathcal D} \to {\mathcal C}$ let $\mathrm {Fun}^{\mathcal C}({\mathcal D},-): \infty \mathrm {Cat}_{/{\mathcal C}} \to \infty \mathrm {Cat}_{/{\mathcal C}}$ be the right adjoint of the functor ${\mathcal D} \times _{\mathcal C} (-): \infty \mathrm {Cat}_{/{\mathcal C}} \to \infty \mathrm {Cat}_{/{\mathcal C}}$ .

Proof. A functor ${\mathcal B} \to {\mathcal A}$ is a cartesian fibration if and only if the pullbacks along any functors $[2] \to {\mathcal A}$ are cartesian fibrations by [Reference Lurie27, Corollary 2.4.2.10.]. A functor ${\mathcal B} \to {\mathcal A}$ is an exponentiable fibration if and only if the pullbacks along any functors $[2] \to {\mathcal A}$ are exponentiable fibrations as a consequence of [Reference Lurie27, Proposition B.3.14.] and Remark 3.11.

So by Remark 3.10 we can reduce to the case that ${\mathcal A}=[2].$ The locally cartesian fibration ${\mathcal P}^{[2]}({\mathcal C}) \to [2]$ classifies three functors ${\mathrm {f}}: {\mathcal P}({\mathcal C}_2) \to {\mathcal P}({\mathcal C}_1), {g}: {\mathcal P}({\mathcal C}_1) \to {\mathcal P}({\mathcal C}_0), {\mathrm {h}}: {\mathcal P}({\mathcal C}_2) \to {\mathcal P}({\mathcal C}_0)$ , and a natural transformation $\alpha : {g} \circ {\mathrm {f}} \to {\mathrm {h}}.$ By [Reference Lurie26, Proposition B.3.2., Proposition B.3.14.] a functor ${\mathcal C} \to [2]$ is an exponentiable fibration if and only if $\alpha $ is an equivalence. By [Reference Lurie27, Proposition 2.4.2.8.] the locally cartesian fibration ${\mathcal P}^{{\mathcal A}}({\mathcal C}) \to {\mathcal A}$ is a cartesian fibration if and only if $\alpha $ is an equivalence.

Proof. We apply Proposition 3.22 and use that every locally cartesian fibration over $[1] $ is a cartesian fibration.

Proof. By Proposition 3.22 the functor $\phi $ is an exponentiable fibration if and only if the locally cartesian fibration ${\mathcal P}^{{\mathcal B}}({\mathcal A}) \to {\mathcal B}$ is a cartesian fibration. By [Reference Lurie27, Proposition 2.4.2.8.] the locally cartesian fibration ${\mathcal P}^{{\mathcal B}}({\mathcal A}) \to {\mathcal B}$ is a cartesian fibration if and only if for every morphisms $\alpha : {\mathrm X} \to {\mathrm Y}, \beta : {\mathrm Y} \to {\mathrm Z} $ the induced natural transformation $\theta : \alpha ^* \circ \beta ^* \to (\beta \circ \alpha )^*: {\mathcal P}({\mathcal A}_{\mathrm Z}) \to {\mathcal P}({\mathcal A}_{\mathrm X})$ is an equivalence. If ${\mathrm X}$ does not belong to ${\mathcal A}$ , the fiber of $\phi $ over ${\mathrm X}$ is empty so that ${\mathcal P}({\mathcal A}_{\mathrm X})$ is contractible. So there is nothing to show and we assume that ${\mathrm X} $ belongs to ${\mathcal A}$ so that ${\mathcal P}({\mathcal A}_{\mathrm X})\simeq {\mathcal S}.$ If ${\mathrm Z}$ does not belong to ${\mathcal A}$ , the fiber of $\phi $ over ${\mathrm Z}$ is empty so that ${\mathcal P}({\mathcal A}_{\mathrm Z})$ is contractible. In this case there is nothing to show, too, because $\alpha ^*,\beta ^*$ preserve small colimits and so the initial object so that $\theta $ is an endomorphism of the initial object in ${\mathcal P}({\mathcal A}_{\mathrm X})\simeq {\mathcal S}$ and so an equivalence. So we can assume that ${\mathrm Z}$ belongs to ${\mathcal A}$ . By assumption we find that ${\mathrm Y}$ belongs to ${\mathcal A}$ . In this case the small colimits preserving functors $\alpha ^*,\beta ^*:{\mathcal S} \to {\mathcal S}$ are the identity on objects and so preserve the final object. This implies that $\alpha ^*,\beta ^*:{\mathcal S} \to {\mathcal S}$ are equivalences and $\theta $ is an equivalence since the component of $\theta $ at the final space is an endomorphism of the final space and so an equivalence.

Proof. The result follows from Corollary 3.24 since for every $ {i} \geq 0$ the ordered subset $[{n}] \cong \{{i}+0,\ldots,{i}+{n} \} \subset [{m}]$ is an interval.

Proof. Let ${\mathcal C} \to {\mathcal A}$ be a locally cartesian fibration of small $\infty $ -categories. By Lemma 3.18 the functor ${\mathcal C} \to {\mathcal A}$ is a cartesian fibration if and only if the functor $ {\mathcal P}^{\mathcal A}({\mathcal C}) \to {\mathcal A}$ is a cartesian fibration. By Proposition 3.22 the functor ${\mathcal C} \to {\mathcal A}$ is an exponentiable fibration if and only if the functor ${\mathcal P}^{{\mathcal A}}({\mathcal C}) \to {\mathcal A}$ is a cartesian fibration.

Notation 3.28. Let

$$ \begin{align*}\mathrm{LoCART} \subset \mathrm{Fun}([1],\infty\mathrm{Cat})\end{align*} $$

be the full subcategory of locally cartesian fibrations.

Notation 3.29. Let

$$ \begin{align*}\mathrm{LoCART}^{\mathrm L} \subset \mathrm{Lo}\widehat{{\mathrm{CART}}}\end{align*} $$

be the subcategory of locally cartesian fibrations whose target is small and whose fibers are presentable and commutative squares

such that for every ${\mathrm A} \in {\mathcal A}$ the induced functor ${\mathcal C}_{\mathrm A} \to {\mathcal D}_{\psi ({\mathrm A})}$ on the fiber over ${\mathrm A}$ preserves small colimits.

Proof. By Proposition 3.16 (2) for every locally cartesian fibration ${\mathcal D} \to {\mathcal B}$ whose fibers admit small colimits, and functor ${\mathcal C} \to {\mathcal A}$ restriction along the embedding ${\mathcal C} \to {\mathcal P}^{\mathcal A}({\mathcal C})$ induces an equivalence

$$ \begin{align*}\Theta({\mathcal P}^{\mathcal A}({\mathcal C}), {\mathcal D})^{\mathrm L} \to \Theta({\mathcal C}, {\mathcal D}).\end{align*} $$

This implies the result by [Reference Heine20, Remark 2.75.] since $\Theta ({\mathcal C}, {\mathcal D})$ is the morphism $\infty $ -category of the $(\infty ,2)$ -category $\mathrm {Fun}([1],\infty \widehat {\mathrm {Cat}})$ and $\Theta ({\mathcal P}^{\mathcal A}({\mathcal C}), {\mathcal D})^{\mathrm L}$ is the morphism $\infty $ -category of the $(\infty ,2)$ -category $\mathrm {LoCART}^{\mathrm L} $ .

Proof. By Proposition 3.16 (2) for every locally cartesian fibration ${\mathcal D} \to {\mathcal A} $ whose fibers admit small colimits, and functor ${\mathcal C} \to {\mathcal A}$ restriction along the embedding ${\mathcal C} \to {\mathcal P}^{\mathcal A}({\mathcal C})$ induces an equivalence

$$ \begin{align*}\Theta({\mathcal P}^{\mathcal A}({\mathcal C}), {\mathcal D})^{\mathrm L} \to \Theta({\mathcal C}, {\mathcal D}).\end{align*} $$

The latter equivalence is over $\mathrm {Fun}({\mathcal A},{\mathcal A})$ and induces on the fiber over the identity of ${\mathcal A}$ the canonical functor

$$ \begin{align*}\mathrm{Fun}_{\mathcal A}^{\mathrm L}({\mathcal P}^{\mathcal A}({\mathcal C}), {\mathcal D}) \to \mathrm{Fun}_{\mathcal A}({\mathcal C}, {\mathcal D}),\end{align*} $$

which therefore also is an equivalence.

This implies the result by [Reference Heine20, Remark 2.75.] since $\mathrm {Fun}_{\mathcal A}({\mathcal C}, {\mathcal D})$ is the morphism $\infty $ -category of the $(\infty ,2)$ -category $\infty \widehat {\mathrm {Cat}}_{/{\mathcal A}}$ and $\mathrm {Fun}_{\mathcal A}^{\mathrm L}({\mathcal P}^{\mathcal A}({\mathcal C}), {\mathcal D}) $ is the morphism $\infty $ -category of the $(\infty ,2)$ -category $ \mathrm {LoCART}^{\mathrm L}_{\mathcal A} $ .

Notation 3.32. Let ${\mathcal L} \subset \mathrm {Fun}([1],\infty \mathrm {Cat})$ be the full subcategory of left fibrations.

Proof. Let ${\mathcal C} \to \infty \mathrm {Cat}$ be a functor classifying a cocartesian fibration ${\mathcal B} \to {\mathcal C}.$ We prove that for any $\infty $ -category ${\mathcal C}$ functors ${\mathcal C} \to {\mathcal L}$ over $\infty \mathrm {Cat}$ naturally correspond to functors ${\mathcal C} \to \mathrm {Fun}^{\infty \mathrm {Cat}}({\mathcal U},{\mathcal S} \times \infty \mathrm {Cat})$ over $\infty \mathrm {Cat}$ . Functors ${\mathcal C} \to \mathrm {Fun}^{\infty \mathrm {Cat}}({\mathcal U},{\mathcal S} \times {\mathcal C})$ over $\infty \mathrm {Cat}$ correspond to sections of the functor $\mathrm {Fun}^{{\mathcal C}}({\mathcal B},{\mathcal S} \times {\mathcal C}) \to {\mathcal C}$ . Such sections correspond to functors ${\mathcal B} \to {\mathcal S} \times {\mathcal C}$ over ${\mathcal C}$ and so to functors ${\mathcal B} \to {\mathcal S}.$ A functor ${\mathcal B} \to {\mathcal L} \subset \mathrm {Fun}([1],\infty \mathrm {Cat})$ over $\infty \mathrm {Cat}$ is classified by a map of cocartesian fibrations ${\mathcal A} \to {\mathcal B}$ over ${\mathcal C}$ that is fiberwise a left fibration and so a left fibration by [Reference Lurie27, Proposition 2.4.2.4., Proposition 2.4.2.11.]. Such a functor corresponds to a left fibration ${\mathcal A} \to {\mathcal B}$ classifying a functor ${\mathcal B} \to {\mathcal S}.$

Proof. By Corollary 3.36 there is a canonical equivalence

$$ \begin{align*}{\mathcal L} \simeq \mathrm{Fun}^{\infty\mathrm{Cat}}({\mathcal U},{\mathcal S} \times \infty\mathrm{Cat})\end{align*} $$

over $\infty \mathrm {Cat}.$ By [Reference Gepner, Haugseng and Nikolaus12, Proposition 7.3.] the functor $\mathrm {Fun}^{\infty \mathrm {Cat}}({\mathcal U},{\mathcal S} \times \infty \mathrm {Cat}) \to \infty \mathrm {Cat} $ classifies the functor

$$ \begin{align*}\mathrm{Fun}(-,\infty\mathrm{Cat}): \infty\mathrm{Cat}^{\mathrm{op}} \to \infty\widehat{\mathrm{Cat}}.\\[-39pt] \end{align*} $$

Notation 3.35. Let ${\mathcal C} \to {\mathcal D}$ be a cocartesian fibration classifying a functor $\alpha : {\mathcal D} \to \infty \mathrm {Cat}.$ We write ${\mathcal C}^{\mathrm {rev}} \to {\mathcal D}$ for the cocartesian fibration classifying the composition

$$ \begin{align*}{\mathcal D} \xrightarrow{\alpha} \infty\mathrm{Cat} \xrightarrow{(-)^{\mathrm{op}}}\infty\mathrm{Cat}.\end{align*} $$

Proof. By Corollary 3.33 there is a canonical equivalence

$$ \begin{align*}{\mathcal L} \simeq \mathrm{Fun}^{\infty\mathrm{Cat}}({\mathcal U},{\mathcal S} \times \infty\mathrm{Cat}) \end{align*} $$

over $\infty \mathrm {Cat}.$ Pulling back the latter equivalence along the equivalence $(-)^{\mathrm {op}}: \infty \mathrm {Cat} \simeq \infty \mathrm {Cat}$ we obtain the desired equivalence

$$ \begin{align*}{\mathcal R} \simeq \mathrm{Fun}^{\infty\mathrm{Cat}}({\mathcal U}^{\mathrm{rev}},{\mathcal S} \times \infty\mathrm{Cat})\end{align*} $$

over $\infty \mathrm {Cat}.$

Proof. The cocartesian fibration ${\mathcal D} \to {\mathcal C}$ classifies the functor $\psi : {\mathcal C} \to \infty \mathrm {Cat}$ . By Corollary 3.36 there is a canonical equivalence

$$ \begin{align*}{\mathcal R} \simeq \mathrm{Fun}^{\infty\mathrm{Cat}}({\mathcal U}^{\mathrm{rev}},{\mathcal S} \times \infty\mathrm{Cat})\end{align*} $$

over $\infty \mathrm {Cat}.$ The pullback of this equivalence along $\psi $ is the desired equivalence

$$ \begin{align*}{\mathcal P}^{\mathcal A}({\mathcal C}) \simeq \mathrm{Fun}^{{\mathcal A}}({\mathcal C}^{\mathrm{rev}},{\mathcal S} \times {\mathcal A})\end{align*} $$

over ${\mathcal A}.$

Proof. By definition the functor ${\mathcal P}^{\mathcal C}({\mathcal D}) \to {\mathcal C} $ is the pullback of $\rho : {\mathcal R} \to \infty \mathrm {Cat}$ along $\psi : {\mathcal C} \to \infty \mathrm {Cat} $ and so a cartesian fibration, which by Corollary 3.34 classifies the composition

$$ \begin{align*}{\mathcal C}^{\mathrm{op}} \xrightarrow{\psi^{\mathrm{op}}} \infty\mathrm{Cat}^{\mathrm{op}} \xrightarrow{\mathrm{Fun}((-)^{\mathrm{op}},\infty\mathrm{Cat})} \infty\widehat{\mathrm{Cat}}.\\[-28pt]\end{align*} $$

Proof. Let ${\mathcal C} \to {\mathcal A},{\mathcal D} \to {\mathcal A}$ be functors of small $\infty $ -categories. Square (4.1) induces on morphism $\infty $ -categories the commutative square

which identifies with the commutative square

(4.2)

where we use Remark 3.10. The bottom and top functors of square (4.2) are fully faithful. Therefore the functor from the left upper $\infty $ -category of square (4.2) to the pullback is fully faithful, too. Consequently, the square (4.2) is a pullback square if the latter functor is essentially surjective. This means that every functor ${\mathcal C} \to {\mathcal P}^{\mathcal A}({\mathcal D})$ over ${\mathcal A}$ whose pullback to $\iota ({\mathcal A})$ lands in ${\mathcal D}$ , already lands in ${\mathcal D}, $ which is clear.

Consequently, the functor

$$ \begin{align*}\theta: \infty\mathrm{Cat}_{/{\mathcal A}} \to \infty\mathrm{Cat}_{/\iota({\mathcal A})} \times_{\mathrm{LoCART}^{\mathrm L}_{\iota({\mathcal A})}} \mathrm{LoCART}^{\mathrm L}_{{\mathcal A}}\end{align*} $$

is fully faithful. So it remains to see that $\theta $ is essentially surjective.

Let ${\mathcal B} \to \iota ({\mathcal A})$ be a functor, $ {\mathcal D} \to {\mathcal A}$ a locally cartesian fibration whose fibers admit small colimits and whose fiber transports preserve small colimits, and $\alpha : {\mathcal P}^{\iota ({\mathcal A})}({\mathcal B}) \simeq \iota ({\mathcal A}) \times _{\mathcal A} {\mathcal D} $ an equivalence over $\iota ({\mathcal A})$ . Let ${\mathcal C} \subset {\mathcal D}$ be the essential image of the Yoneda-embedding ${\mathcal B} \subset {\mathcal P}^{\iota ({\mathcal A})}({\mathcal B}) \simeq \iota ({\mathcal A}) \times _{\mathcal A} {\mathcal D} \subset {\mathcal D}$ . Then there is a canonical equivalence $\iota ({\mathcal A}) \times _{\mathcal A}{\mathcal C} \simeq {\mathcal B}$ over $\iota ({\mathcal A})$ and the embedding ${\mathcal C} \subset {\mathcal D}$ over ${\mathcal A}$ extends to a canonical functor $\beta : {\mathcal P}^{\mathcal A}({\mathcal C}) \to {\mathcal D}$ over ${\mathcal A},$ whose pullback to $\iota ({\mathcal A})$ is the equivalence $\alpha : \iota ({\mathcal A}) \times _{\mathcal A}{\mathcal P}^{\mathcal A}({\mathcal C}) \simeq {\mathcal P}^{\iota ({\mathcal A})}({\mathcal B}) \simeq \iota ({\mathcal A}) \times _{\mathcal A}{\mathcal D}.$ Therefore it is enough to see that $\beta $ is an equivalence. For this by Remark 3.10 we can reduce to the case that ${\mathcal A}=[1]$ because a functor over ${\mathcal A}$ is an equivalence if all pullbacks along functors $[1] \to {\mathcal A}$ are equivalences. Since $\beta $ is essentially surjective, we need to check that for any ${\mathrm X} \in {\mathcal P}({\mathcal C}_0), {\mathrm Y} \in {\mathcal P}({\mathcal C}_1)$ the canonical map $\theta _{{\mathrm X},{\mathrm Y}}: {\mathcal P}^{[1]}({\mathcal C})({\mathrm X}, {\mathrm Y}) \to {\mathcal D}(\beta ({\mathrm X}),\beta ({\mathrm Y}))$ induced by $\beta $ is an equivalence. Let ${\mathrm G}: {\mathcal P}({\mathcal C}_1) \to {\mathcal P}({\mathcal C}_0)$ be the left adjoint functor classified by the cartesian fibration ${\mathcal P}^{[1]}({\mathcal C}) \to [1]$ and ${\mathrm G}': {\mathcal D}_1 \to {\mathcal D}_0$ the left adjoint functor classified by the cartesian fibration ${\mathcal D} \to [1]$ . The unique cartesian lift ${\mathrm G}({\mathrm Y}) \to {\mathrm Y}$ , where ${\mathrm G}({\mathrm Y}) \in {\mathcal P}({\mathcal C}_0)$ , is sent to a morphism $\beta ({\mathrm G}({\mathrm Y})) \to \beta ({\mathrm Y})$ that factors as $\beta ({\mathrm G}({\mathrm Y})) \xrightarrow {\rho _{\mathrm Y}} {\mathrm G}'(\beta ({\mathrm Y})) \to \beta ({\mathrm Y})$ , where ${\mathrm G}'(\beta ({\mathrm Y})) \in {\mathcal D}_0$ . The map $\theta _{{\mathrm X},{\mathrm Y}}$ identifies with the map

$$ \begin{align*}{\mathcal P}({\mathcal C}_0)({\mathrm X}, {\mathrm G}({\mathrm Y})) \to {\mathcal D}_0(\beta({\mathrm X}), \beta({\mathrm G}({\mathrm Y}))) \to {\mathcal D}_0(\beta({\mathrm X}), {\mathrm G}'(\beta({\mathrm Y}))).\end{align*} $$

Since the first map is an equivalence, it is enough to see that $\rho _{\mathrm Y}$ is an equivalence for every ${\mathrm Y} \in {\mathcal P}({\mathcal C}_1).$ Since $\beta _{\mathcal C}$ is fully faithful, $\theta _{{\mathrm X},{\mathrm Y}}$ is an equivalence if ${\mathrm X} \in {\mathcal C}_0, {\mathrm Y} \in {\mathcal C}_1$ . Because ${\mathcal P}({\mathcal C}_0)$ is generated by ${\mathcal C}_0$ under small colimits, $\theta _{{\mathrm X},{\mathrm Y}}$ is an equivalence for all ${\mathrm X} \in {\mathcal P}({\mathcal C}_0), {\mathrm Y} \in {\mathcal C}_1$ . Thus $\rho _{\mathrm Y}$ is an equivalence for every ${\mathrm Y} \in {\mathcal C}_1$ . Therefore $\rho _{\mathrm Y}$ is also an equivalence for every ${\mathrm Y} \in {\mathcal P}({\mathcal C}_1)$ because ${\mathcal P}({\mathcal C}_1)$ is generated by ${\mathcal C}_1$ under small colimits and the functors ${\mathrm G}, {\mathrm G}'$ preserve small colimits and $\rho _{\mathrm Y}$ is natural in ${\mathrm Y}$ .

Proof. Square (4.3) induces on the fiber over every small $\infty $ -category ${\mathcal A}$ the pullback square (4.1) of Theorem 4.1, and therefore itself is a pullback square.

Notation 4.2. Let

• ○ $\mathrm {EXP}\subset \mathrm {Fun}([1],\infty \mathrm {Cat})$ be the full subcategory of exponentiable fibrations,

• ○ $\mathrm {CART} \subset \mathrm {Fun}([1],\infty \mathrm {Cat}) $ be the full subcategory of cartesian fibrations,

• ○ ${\mathrm {CART}}^{\mathrm L} \subset \mathrm {LoCART}^{\mathrm L} $ be the full subcategory of cartesian fibrations.

Proof. By Corollary 4.1 the canonical commutative square (4.3) is a pullback square. The square (4.4) embeds into the commutative square (4.3), and via Proposition 3.22 is a pullback square, too.

Proof. By Theorem 4.1 there is a canonical pullback square of $\infty $ -categories:

Consequently, it is enough to construct an equivalence between the pullbacks

and

We construct this equivalence by giving for every $\infty $ -category ${\mathcal A}$ a bijection between equivalence classes of objects of the $\infty $ -categories $ \mathrm {Fun}({\mathcal A},{\mathcal A}), \mathrm {Fun}({\mathcal A},{\mathcal B})$ natural in ${\mathcal A}.$ A functor ${\mathcal A} \to {\mathcal B}$ is classified by a pair of cocartesian fibrations ${\mathcal X} \to {\mathcal A}, {\mathcal Y} \to {\mathcal A}$ and a map of cocartesian fibrations ${\mathcal M} \to {\mathcal A} \times [1]$ over ${\mathcal A}$ that induces on the fiber over every object ${\mathrm A}\in {\mathcal A}$ a cartesian fibration ${\mathcal M}_{\mathrm A} \to [1]$ classifying a small colimits preserving functor ${\mathcal M}_{\mathrm A}^1 \to {\mathcal M}_{\mathrm A}^0$ and such that there are equivalences ${\mathcal M}_0 \simeq {\mathcal P}^{\mathcal A}({\mathcal X}), {\mathcal M}_1 \simeq {\mathcal P}^{\mathcal A}({\mathcal Y})$ over ${\mathcal A}$ . By [Reference Heine14, Lemma 2.44.] a functor ${\mathcal W} \to {\mathcal A} \times [1]$ is a map of cocartesian fibrations over ${\mathcal A}$ that induces on the fiber over every object ${\mathrm A}\in {\mathcal A}$ a cartesian fibration over $ [1]$ if and only if it is a map of cartesian fibrations over $[1]$ that induces on the fiber over every object of $[1]$ a cocartesian fibration over ${\mathcal A}$ . Consequently, a functor ${\mathcal A} \to {\mathcal B}$ is likewise classified by a pair of cocartesian fibrations ${\mathcal X} \to {\mathcal A}, {\mathcal Y} \to {\mathcal A}$ and a map of cartesian fibrations ${\mathcal M} \to {\mathcal A} \times [1]$ over $[1]$ that induces on the fibers over $0,1 \in [1]$ the cocartesian fibrations ${\mathcal P}^{\mathcal A}({\mathcal X}), {\mathcal P}^{\mathcal A}({\mathcal Y})$ , respectively, and induces on the fiber over every object ${\mathrm A}\in {\mathcal A}$ a cartesian fibration ${\mathcal M}_{\mathrm A} \to [1]$ classifying a small colimits preserving functor ${\mathcal M}_{\mathrm A}^1 \to {\mathcal M}_{\mathrm A}^0$ . In other words, a functor ${\mathcal A} \to {\mathcal B}$ is classified by a pair of cocartesian fibrations ${\mathcal X} \to {\mathcal A}, {\mathcal Y} \to {\mathcal A}$ and a functor ${\mathcal P}^{\mathcal A}({\mathcal Y}) \to {\mathcal P}^{\mathcal A}({\mathcal X})$ over ${\mathcal A}$ that induces on the fiber over every object of ${\mathcal A}$ a small colimits preserving functor. By Proposition 3.16 the latter is uniquely determined by its restriction, a functor ${\mathcal Y} \to {\mathcal P}^{\mathcal A}({\mathcal X})$ over ${\mathcal A}$ , which corresponds by Proposition 3.37 to a functor ${\mathcal X}^{\mathrm {rev}} \times _{\mathcal A} {\mathcal Y} \to {\mathcal S}$ . The latter is classified by a left fibration ${\mathcal A} \to {\mathcal X}^{\mathrm {rev}} \times _{\mathcal A} {\mathcal Y}$ or equivalently by a map of cocartesian fibrations ${\mathcal A} \to {\mathcal X}^{\mathrm {rev}} \times _{\mathcal A} {\mathcal Y} $ over ${\mathcal A}$ that induces on the fiber over every object of ${\mathcal A}$ a left fibration. So a functor ${\mathcal A} \to {\mathcal B}$ is classified by a pair of cocartesian fibrations ${\mathcal X} \to {\mathcal A}, {\mathcal Y} \to {\mathcal A}$ and a map of cocartesian fibrations ${\mathcal A} \to {\mathcal X}^{\mathrm {rev}} \times _{\mathcal A} {\mathcal Y} $ over ${\mathcal A}$ that induces on the fiber over every object of ${\mathcal A}$ a left fibration. The latter is precisely classified by a functor ${\mathcal A} \to {\mathcal A}.$

Notation 5.2. Let ${\mathrm {Cocart}} \subset \mathrm {Fun}([1],\infty \mathrm {Cat})$ be the full subcategory of cocartesian fibrations.

Proof. By Theorem 5.1 there is a canonical equivalence

$$ \begin{align*}\{({\mathcal C},{\mathcal D})\}\times_{\infty\mathrm{Cat} \times \infty\mathrm{Cat}} \mathrm{Cat}_{/[1]} \simeq \{{\mathcal C}^{\mathrm{op}} \times {\mathcal D} \}\times_{\infty\mathrm{Cat}} {\mathcal L} \simeq \mathrm{Fun}({\mathcal C}^{\mathrm{op}}\times{\mathcal D},{\mathcal S}) \simeq \mathrm{Fun}({\mathcal C}^{\mathrm{op}}, \mathrm{Fun}({\mathcal D},{\mathcal S}))\end{align*} $$

that sends a functor ${\mathcal M} \to [1]$ and ${\mathcal M}_0 \simeq {\mathcal C}, {\mathcal M}_1 \simeq {\mathcal D}$ to the functor ${\mathcal C}^{\mathrm {op}} \times {\mathcal D} \subset {\mathcal M}^{\mathrm {op}} \times {\mathcal M} \xrightarrow {{\mathcal M}(-,-)} {\mathcal S}.$

The latter equivalence restricts to an equivalence

$$ \begin{align*}\{({\mathcal C},{\mathcal D})\}\times_{\infty\mathrm{Cat} \times \infty\mathrm{Cat}} {\mathrm{Cocart}}_{[1]} \simeq \mathrm{Fun}({\mathcal C}^{\mathrm{op}}, {\mathcal D}^{\mathrm{op}}) \simeq \mathrm{Fun}({\mathcal C},{\mathcal D})^{\mathrm{op}}\end{align*} $$

that sends a functor ${\mathcal M} \to [1]$ and ${\mathcal M}_0 \simeq {\mathcal C}, {\mathcal M}_1 \simeq {\mathcal D}$ to the functor ${\mathcal C} \to {\mathcal D}$ classified by ${\mathcal M} \to [1]$ .

For every cocartesian fibration ${\mathcal M} \to [1]$ and ${\mathcal M}_0 \simeq {\mathcal C}, {\mathcal M}_1 \simeq {\mathcal D}$ classifying a functor $\alpha : {\mathcal C} \to {\mathcal D}$ there is a canonical equivalence between the functor ${\mathcal C}^{\mathrm {op}} \times {\mathcal D} \subset {\mathcal M}^{\mathrm {op}} \times {\mathcal M} \xrightarrow {{\mathcal M}(-,-)} {\mathcal S}$ and the functor ${\mathcal C}^{\mathrm {op}} \times {\mathcal D} \xrightarrow {\alpha ^{\mathrm {op}} \times {\mathcal D}} {\mathcal D}^{\mathrm {op}} \times {\mathcal D} \xrightarrow {{\mathcal D}(-,-)} {\mathcal S}$ .

Proof. For every $({\mathrm A}, {\mathrm {f}}: {\mathrm X} \to \alpha ({\mathrm A})), ({\mathrm B}, {g}: {\mathrm Y} \to \alpha ({\mathrm B}))\in {\mathcal A} \times _{{\mathcal V}} {\mathcal W}, ({\mathrm C}, {\mathrm {h}}: {\mathrm Z} \to \alpha ({\mathrm C}))\in {\mathcal A} \times _{{\mathcal V}} {\mathcal W} $ the canonical morphism

$$ \begin{align*}({\mathrm{Mor}}_{\mathcal A}({\mathrm A},{\mathrm B}), \alpha({\mathrm{Mor}}_{\mathcal A}({\mathrm A},{\mathrm B})) \times_{{\mathrm{Mor}}_{\mathcal V}({\mathrm X}, \alpha({\mathrm B}))} {\mathrm{Mor}}_{\mathcal V}({\mathrm X},{\mathrm Y}) \otimes ({\mathrm A}, {\mathrm{f}}: {\mathrm X} \to \alpha({\mathrm A})) =\end{align*} $$

$$ \begin{align*}({\mathrm{Mor}}_{\mathcal A}({\mathrm A},{\mathrm B})\otimes{\mathrm A}, \alpha({\mathrm{Mor}}_{\mathcal A}({\mathrm A},{\mathrm B})\otimes {\mathrm A}) \times_{{\mathrm{Mor}}_{\mathcal V}({\mathrm X}, \alpha({\mathrm B}))\otimes {\mathrm X}} {\mathrm{Mor}}_{\mathcal V}({\mathrm X},{\mathrm Y}) \otimes {\mathrm X}) \to ({\mathrm B}, {g}: {\mathrm Y} \to \alpha({\mathrm B}))\end{align*} $$

in ${\mathcal A} \times _{{\mathcal V}} {\mathcal W}$ induces a map

$$ \begin{align*}{\mathcal A} \times_{{\mathcal V}} {\mathcal W}(({\mathrm A}, {\mathrm{f}}: {\mathrm X} \to \alpha({\mathrm A})),\end{align*} $$

$$ \begin{align*}({\mathrm{Mor}}_{\mathcal A}({\mathrm B},{\mathrm C}), \alpha({\mathrm{Mor}}_{\mathcal A}({\mathrm B},{\mathrm C})) \times_{{\mathrm{Mor}}_{\mathcal V}({\mathrm Y}, \alpha({\mathrm C}))} {\mathrm{Mor}}_{\mathcal V}({\mathrm Y},{\mathrm Z}) \to \alpha({\mathrm{Mor}}_{\mathcal A}({\mathrm B},{\mathrm C}))) ) \simeq\end{align*} $$

$$ \begin{align*}{\mathcal A} \times_{{\mathcal V}} {\mathcal W}(({\mathrm A}, {\mathrm{f}}: {\mathrm X} \to \alpha({\mathrm A})) \otimes ({\mathrm B}, {g}: {\mathrm Y} \to \alpha({\mathrm B})),({\mathrm C}, {\mathrm{h}}: {\mathrm Z} \to \alpha({\mathrm C})))=\end{align*} $$

$$ \begin{align*}{\mathcal A} \times_{{\mathcal V}} {\mathcal W}(({\mathrm A} \otimes {\mathrm B}, {\mathrm{f}} \otimes {g}: {\mathrm X} \otimes {\mathrm Y} \to \alpha({\mathrm A} \otimes {\mathrm B})),({\mathrm C}, {\mathrm{h}}: {\mathrm Z} \to \alpha({\mathrm C})))\end{align*} $$

over ${\mathcal A}({\mathrm A}, {\mathrm {Mor}}_{\mathcal A}({\mathrm B},{\mathrm C})) \simeq {\mathcal A}({\mathrm A} \otimes {\mathrm B},{\mathrm C})$ that induces on the fiber over every morphism $\theta : {\mathrm A} \to {\mathrm {Mor}}_{\mathcal A}({\mathrm B},{\mathrm C})$ corresponding to a morphism $\theta ': {\mathrm A} \otimes {\mathrm B} \to {\mathrm C}$ the canonical equivalence

$$ \begin{align*}{\mathcal V}_{/{\mathrm{Mor}}_{\mathcal V}({\mathrm Y}, \alpha({\mathrm C}))}({\mathrm X}, {\mathrm{Mor}}_{\mathcal V}({\mathrm Y},{\mathrm Z})) \simeq {\mathcal V}_{/\alpha({\mathrm C})}({\mathrm X}\otimes{\mathrm Y},{\mathrm Z}).\\[-37pt] \end{align*} $$

Proof. We apply Lemma 6.1 to ${\mathcal V}:=\infty \mathrm {Cat}$ and ${\mathcal W}:= {\mathcal L} \subset \mathrm {Fun}([1], {\mathcal V}).$

Notation 6.3. For every functor ${\mathcal M} \to [1]$ let $\widetilde {{\mathrm {Tw}}}({\mathcal M})$ be the pullback

$$ \begin{align*}{\mathcal M}_0^{\mathrm{op}} \times {\mathcal M}_1 \times_{{\mathcal M}^{\mathrm{op}}\times{\mathcal M}}{\mathrm{Tw}}({\mathcal M}) \to {\mathcal M}_0^{\mathrm{op}} \times {\mathcal M}_1.\end{align*} $$

Proof. By Theorem 5.1 there is a canonical equivalence $\theta $ between $\infty \mathrm {Cat}_{/[1]}$ and the pullback

Consequently, the internal hom of $\infty \mathrm {Cat}_{/[1]}$ corresponds under $\theta $ to the internal hom of ${\mathcal Q}$ for the cartesian monoidal structures. The description of $\theta $ of Theorem 5.1 and the description of the internal hom of Corollary 6.2 imply the result.

Proof. There is a canonical equivalence

$$ \begin{align*}\{\alpha\}\times_{{\mathcal C}({\mathrm X},{\mathrm Y})} \mathrm{Fun}^{\mathcal C}({\mathcal A},{\mathcal B})({\mathrm F},{\mathrm G}) \simeq ([1]\times_ {\mathcal C} \mathrm{Fun}^{\mathcal C}({\mathcal A},{\mathcal B}))({\mathrm F},{\mathrm G}) \simeq\end{align*} $$

$$ \begin{align*}\mathrm{Fun}^{[1]}([1]\times_ {\mathcal C} {\mathcal A},[1]\times_ {\mathcal C} {\mathcal B})({\mathrm F},{\mathrm G}) \simeq\end{align*} $$

$$ \begin{align*}\lim(\widetilde{{\mathrm{Tw}}}([1]\times_ {\mathcal C} {\mathcal A}) \to {\mathcal A}_{\mathrm X}^{\mathrm{op}} \times {\mathcal A}_{\mathrm Y} \xrightarrow{{\mathrm F}^{\mathrm{op}} \times {\mathrm G}} {\mathcal B}_{\mathrm X}^{\mathrm{op}} \times {\mathcal B}_{\mathrm Y} \to ([1]\times_ {\mathcal C} {\mathcal B})^{\mathrm{op}} \times ([1]\times_ {\mathcal C} {\mathcal B}) \xrightarrow{([1]\times_ {\mathcal C} {\mathcal B})(-,-)} {\mathcal S}).\end{align*} $$

Proof. We apply Theorem 6.2 to the functors ${\mathcal M} := {\mathcal C} \times [1] \to [1], {\mathcal N}:= {\mathcal D} \times [1] \to [1] $ and use Remark 6.1.

Proof. By the Grothendieck construction there is a canonical equivalence

$$ \begin{align*}\infty\mathrm{Cat}^{\mathrm{cart}}_{/\Delta} \simeq \mathrm{Fun}(\Delta^{\mathrm{op}}, \infty\mathrm{Cat}).\end{align*} $$

The functor $\mathrm {Fun}(\Delta ^{\mathrm {op}}, \infty \mathrm {Cat}) \simeq \infty \mathrm {Cat}^{\mathrm {cart}}_{/\Delta } \xrightarrow {\rho } \infty \mathrm {Cat}$ is the functor evaluating at $[0]$ . By [Reference Lurie27, Corollary 4.3.2.14., Proposition 4.3.2.17.] this functor admits a right adjoint $ \gamma $ that sends any $\infty $ -category ${\mathcal C}$ to the right Kan extension of the functor $ [0] \xrightarrow {{\mathcal C}} \infty \mathrm {Cat}$ along the embedding $ \{[0] \} \subset \Delta ^{\mathrm {op}}.$ By [Reference Lurie27, Lemma 4.3.2.13., Corollary 4.3.2.14., Proposition 4.3.2.17.] this right Kan extension $\gamma ({\mathcal C}) $ sends any $[{n}] \in \Delta $ to the limit of the functor $ \{0,\ldots,{n} \} \simeq (\{[0] \} \times _{\Delta } \Delta _{/[{n}]})^{\mathrm {op}} \to [0] \xrightarrow {{\mathcal C}} \infty \mathrm {Cat}, $ which identifies with $\mathrm {Fun}(\{0,\ldots,{n} \},{\mathcal C}) \simeq {\mathcal C}^{\times {n}+1}. $ Hence $\gamma ({\mathcal C}) $ is the functor

$$ \begin{align*}\Delta^{\mathrm{op}} \to \infty\mathrm{Cat}, [{n}] \mapsto \mathrm{Fun}(\{0,\ldots,{n} \},{\mathcal C}).\end{align*} $$

This implies the result by definition of $\Delta _{/{\mathcal C}} \to \Delta. $

Notation 7.3. Let ${\mathcal A} \to \Delta , {\mathcal B} \to \Delta $ be double $\infty $ -categories. We write

$$ \begin{align*}{\mathrm{FUN}}({\mathcal A},{\mathcal B}) \subset \mathrm{Fun}_{\Delta}({\mathcal A},{\mathcal B})\end{align*} $$

for the full subcategory spanned by the functors over $\Delta $ preserving cocartesian lifts of morphisms of $\Delta .$

Notation 7.5. We define the following $\infty $ -categories by the following pullback squares:

Construction 7.6. Let ${\mathrm H}: {\mathrm K} \to \infty \mathrm {Cat}$ be a functor that admits a colimit ${\mathcal A}$ . The canonical $\infty \mathrm {Cat}$ -linear functor $\mathrm {Cat}_{\infty /{\mathcal A}} \to \lim _{{k} \in {\mathrm K}}\mathrm {Cat}_{\infty /{\mathrm H}({k})}$ induces on morphism $\infty $ -categories between two functors ${\mathcal C} \to {\mathcal A},{\mathcal D} \to {\mathcal A}$ a functor

(7.1) $$ \begin{align} \theta: \mathrm{Fun}_{{\mathcal A}}({\mathcal C},{\mathcal D}) \to \lim_{{k} \in {\mathrm K}} \mathrm{Fun}_{{\mathrm H}({k})}({\mathrm H}({k}) \times_{{\mathcal A}} {\mathcal C},{\mathrm H}({k}) \times_{{\mathcal A}}{\mathcal D}). \end{align} $$

Proof. Since ${\mathcal C} \to {\mathcal A}$ is an exponentiable fibration, the functor $(-) \times _{{\mathcal A}}{\mathcal C} : \mathrm {Cat}_{\infty /{\mathcal A}} \to \mathrm {Cat}_{\infty /{\mathcal C}} \to \mathrm {Cat}_{\infty /{\mathcal A}}$ preserves small colimits. Hence the canonical functor

$$ \begin{align*}\mathrm{colim}_{{k} \in {\mathrm K}}({\mathrm H}({k}) \times_{{\mathcal A}} {\mathcal C}) \to {\mathcal C}\end{align*} $$

over $ \mathrm {colim} ({\mathrm H}) \simeq {\mathcal A} $ is an equivalence. Since the functor $\mathrm {Fun}_{{\mathcal A}}(-,{\mathcal D}): \mathrm {Cat}_{\infty /{\mathcal A}}^{\mathrm {op}} \to \infty \mathrm {Cat}$ preserves limits, the following canonical functor is an equivalence:

$$ \begin{align*}\mathrm{Fun}_{{\mathcal A}}({\mathcal C},{\mathcal D}) \to \mathrm{Fun}_{{\mathcal A}}(\mathrm{colim}_{{k} \in {\mathrm K}}{\mathrm H}({k}) \times_{{\mathcal A}} {\mathcal C},{\mathcal D}) \simeq \lim_{{k} \in {\mathrm K}} \mathrm{Fun}_{{\mathrm H}({k})}({\mathrm H}({k}) \times_{{\mathcal A}} {\mathcal C},{\mathrm H}({k}) \times_{{\mathcal A}}{\mathcal D}).\\[-47pt] \end{align*} $$

Proof. There is a commutative square:

By Lemma 7.7 both horizontal functors in this square are equivalences. This implies the result.

Notation 7.9. Let ${n} \geq 0$ and $\phi : {\mathcal C} \to [{n}]$ be a functor. Pulling back diagram (7) along $\phi $ we obtain the following diagram over ${\mathcal C}:$

We write ${\mathcal C}'$ for the colimit of the latter diagram.

Proof. By Corollary 3.26 for every $1 \leq {i} \leq {n}$ the inert maps $[1]\cong \{{i}-1< {i}\}\subset [{n}]$ are exponentiable fibrations. Hence the pullback of the canonical functor ${\mathcal C}' \to {\mathcal C}$ over $[{n}]$ along the map $[1]\cong \{{i}-1< {i}\}\subset [{n}]$ identifies with the canonical equivalence

$$ \begin{align*} \emptyset \coprod_{\emptyset} (\{{i}-1<{i}\}\times_{[{n}]} {\mathcal C}) \coprod_{\emptyset} \emptyset \to \{{i}-1<{i}\}\times_{[{n}]} {\mathcal C}.\end{align*} $$

This implies (2) by Corollary 7.8.

We prove (1). Since the pullback of the canonical functor ${\mathcal C}' \to {\mathcal C}$ over $[{n}]$ along the map $[1]\cong \{{i}-1< {i}\}\subset [{n}]$ is an equivalence, the functor ${\mathcal C}' \to [{n}]$ is the colimit of the following diagram:

This implies (1) by [Reference Ayala and Francis2, Corollary 2.2.13., Corollary 2.2.11.].

Proof. By 7.10 for every functor ${\mathcal C} \to [{n}]$ the functor ${\mathcal C}' \to [{n}]$ is an exponentiable fibration, and for every exponentiable fibration ${\mathcal D} \to [{n}]$ the induced functor

$$ \begin{align*}\mathrm{Fun}_{[{n}]}({\mathcal D},{\mathcal C}') \to \mathrm{Fun}_{[{n}]}({\mathcal D},{\mathcal C})\end{align*} $$

is an equivalence. This guarantees that the embedding $\mathrm {EXP}_{[{n}]} \subset \infty \mathrm {Cat}_{/[{n}]}$ admits a right adjoint that sends any functor ${\mathcal C} \to [{n}]$ to the functor ${\mathcal C}' \to [{n}]$ .

Proof. By Yoneda the cartesian fibration $\Delta \times _{\infty \mathrm {Cat}} {\mathrm {Cart}}=\tau ^*\infty {\mathrm {CAT}} \to \Delta $ classifies the functor

$$ \begin{align*}{\mathrm{FUN}}({\mathrm N}_{\mid \Delta},\tau^*\infty{\mathrm{CAT}}) \simeq {\mathrm{FUN}}(\tau^*{\mathrm N}_{\mid \Delta},\infty{\mathrm{CAT}}): \Delta^{\mathrm{op}} \to \infty\widehat{\mathrm{Cat}}.\end{align*} $$

Since the full subcategory $\Delta \subset \infty \mathrm {Cat}$ is dense, it is enough to verify that the functor $ \infty \mathrm {Cat}^{\mathrm {op}} \to \infty \widehat {\mathrm {Cat}}$ classified by the cartesian fibration ${\mathrm {CART}} \to \infty \mathrm {Cat}$ and the functor

$$ \begin{align*}{\mathrm{FUN}}(\tau^*{\mathrm N},\infty{\mathrm{CAT}}): \infty\mathrm{Cat}^{\mathrm{op}} \to \infty\widehat{\mathrm{Cat}}\end{align*} $$

send for every $\infty $ -category ${\mathcal C}$ the colimit of the functor $\rho : \Delta \times _{\infty \mathrm {Cat}} \infty \mathrm {Cat}_{/{\mathcal C}} \to \Delta \subset \infty \mathrm {Cat}$ to a limit. We first prove the second statement: because the functor

$$ \begin{align*}{\mathrm{FUN}}(-,\infty\mathrm{Cat}): \mathrm{Seg}(\infty\mathrm{Cat})^{\mathrm{op}} \to \infty\mathrm{Cat}\end{align*} $$

preserves small limits, it suffices to show that the embedding ${\mathrm N}: \infty \mathrm {Cat} \to \mathrm {Seg}({\mathcal S})\subset \mathrm {Seg}(\infty \mathrm {Cat})$ preserves the colimit of the functor $\Delta \times _{\infty \mathrm {Cat}} \infty \mathrm {Cat}_{/{\mathcal C}} \to \Delta \subset \infty \mathrm {Cat}$ . The embedding ${\mathrm N}: \infty \mathrm {Cat} \to \mathrm {Seg}({\mathcal S})$ preserves this colimit because $\mathrm {Seg}({\mathcal S})$ is a localization of ${\mathcal P}(\Delta )$ and so for every $\infty $ -category ${\mathcal C}$ the Segal space ${\mathrm N}({\mathcal C}) $ is the colimit of the functor

$$ \begin{align*}\Delta \times_{\mathrm{Seg}({\mathcal S})} \mathrm{Seg}({\mathcal S})_{/{\mathrm N}({\mathcal C})} \to \Delta \subset \infty\mathrm{Cat} \subset \mathrm{Seg}({\mathcal S}),\end{align*} $$

which factors as ${\mathrm N} \rho .$ The embedding $ \mathrm {Seg}({\mathcal S}) \subset \mathrm {Seg}(\infty \mathrm {Cat})$ preserves small colimits since the embedding $\mathrm {Fun}(\Delta ^{\mathrm {op}},{\mathcal S}) \subset \mathrm {Fun}(\Delta ^{\mathrm {op}},\infty \mathrm {Cat})$ preserve small colimits and Segal equivalences: the generating Segal equivalences for $\mathrm {Fun}(\Delta ^{\mathrm {op}},{\mathcal S})$ are the canonical morphisms

$$ \begin{align*}\Delta^1 \coprod_{\Delta^0} \ldots \coprod_{\Delta^0} \Delta^1 \to \Delta^{n}\end{align*} $$

for ${n} \geq 0$ , while the generating Segal equivalences for $\mathrm {Fun}(\Delta ^{\mathrm {op}},\infty \mathrm {Cat})$ are the canonical morphisms

$$ \begin{align*}\Delta^1 \coprod_{\Delta^0} \ldots \coprod_{\Delta^0} \Delta^1 \times {\mathcal A} \to \Delta^{n} \times {\mathcal A}\end{align*} $$

for ${n} \geq 0$ and ${\mathcal A}$ a small $\infty $ -category.

We finish the proof by showing that the cartesian fibration ${\mathrm {CART}} \to \infty \mathrm {Cat}$ classifies a small limits preserving functor $ \infty \mathrm {Cat}^{\mathrm {op}} \to \infty \widehat {\mathrm {Cat}}$ . Let ${\mathrm H}: {\mathrm K} \to \infty \mathrm {Cat}$ be a functor that admits a colimit ${\mathcal A}$ . The canonical $\infty \mathrm {Cat}$ -linear functor $\rho : {\mathrm {CART}}_{{\mathcal A}} \to \lim _{{k} \in {\mathrm K}}{\mathrm {CART}}_{{\mathrm H}({k})}$ induces on morphism $\infty $ -categories between two functors ${\mathcal C} \to {\mathcal A},{\mathcal D} \to {\mathcal A}$ the functor (7.1). The latter is an equivalence by Lemma 7.7 because by Corollary 3.27 every cartesian fibration is an exponentiable fibration. The induced map $\iota (\rho )$ is essentially surjective because it identifies with the canonical equivalence

$$ \begin{align*}\iota(\mathrm{Fun}({\mathcal A},\infty\mathrm{Cat}^{\mathrm{op}})) \to \lim_{{k} \in {\mathrm K}}\iota(\mathrm{Fun}({\mathrm H}({k}),\infty\mathrm{Cat}^{\mathrm{op}})).\end{align*} $$

Proof. By Proposition 7.12 the cartesian fibration $\widehat {{\mathrm {CART}}} \to \infty \widehat {\mathrm {Cat}}$ classifies the functor

$$ \begin{align*}{\mathrm{FUN}}(\tau^*{\mathrm N},\infty\widehat{\mathrm{Cat}}): \infty\widehat{\mathrm{Cat}}^{\mathrm{op}} \to \infty\widehat{\widehat{\mathrm{Cat}}}.\end{align*} $$

Hence the cartesian subfibration ${\mathrm {CART}}^{\mathrm L} \to \infty \widehat {\mathrm {Cat}}$ of $ \widehat {{\mathrm {CART}}} \to \infty \widehat {\mathrm {Cat}}$ classifies the subfunctor

$$ \begin{align*}{\mathrm{FUN}}(\tau^*{\mathrm N},\mathrm{Pr}^{\mathrm L}): \infty\widehat{\mathrm{Cat}}^{\mathrm{op}} \to \infty\widehat{\widehat{\mathrm{Cat}}}.\\[-37pt] \end{align*} $$

Proof. We first prove (2). By Corollary 4.3 there is a canonical pullback square

(7.2)

of cartesian fibrations over $\infty \mathrm {Cat}$ . Taking the pullback of the latter square along the embedding $\Delta \subset \infty \mathrm {Cat}$ gives a pullback square

of cartesian fibrations over $\Delta $ . Since the full subcategory of double $\infty $ -categories within the $\infty $ -category of cartesian fibrations over $\Delta $ is closed under small limits, the latter pullback square is a pullback square of double $\infty $ -categories after we have proven that the cartesian fibrations $\infty {\mathrm {CAT}} \to \Delta , \mathrm {PR}^{\mathrm L}\to \Delta $ are double $\infty $ -categories. This will then also imply (1).

So it remains to prove that the cartesian fibrations $\infty {\mathrm {CAT}} \to \Delta , \mathrm {PR}^{\mathrm L}\to \Delta $ are double $\infty $ -categories.

The cartesian fibration $\infty {\mathrm {CAT}} \to \Delta $ is a double $\infty $ -category if for every ${n} \geq 0$ the induced functor

(7.3) $$ \begin{align} \infty{\mathrm{CAT}}_{[{n}]} \to \infty{\mathrm{CAT}}_{[1]} \times_{\infty{\mathrm{CAT}}_{[0]}} \ldots \times_{\infty{\mathrm{CAT}}_{[0]}} \infty{\mathrm{CAT}}_{[1]} \end{align} $$

is an equivalence. If this is shown, passing to a larger universe also the functor

$$ \begin{align*}\infty\widehat{{\mathrm{CAT}}}_{[{n}]} \to \infty\widehat{{\mathrm{CAT}}}_{[1]} \times_{\infty\widehat{{\mathrm{CAT}}}_{[0]}} \ldots \times_{\infty\widehat{{\mathrm{CAT}}}_{[0]}} \infty\widehat{{\mathrm{CAT}}}_{[1]}\end{align*} $$

is an equivalence, which by definition of $\mathrm {Pr}^{\mathrm L}$ restricts to an equivalence

$$ \begin{align*}\mathrm{Pr}^{\mathrm L}_{[{n}]} \to \mathrm{Pr}^{\mathrm L}_{[1]} \times_{\mathrm{Pr}^{\mathrm L}_{[0]}} \ldots \times_{\mathrm{Pr}^{\mathrm L}_{[0]}} \mathrm{Pr}^{\mathrm L}_{[1]}.\end{align*} $$

Therefore the cartesian fibration $\mathrm {PR}^{\mathrm L}\to \Delta $ is a double $\infty $ -category if the cartesian fibration $\infty {\mathrm {CAT}} \to \Delta $ is a double $\infty $ -category.

So it remains to prove that the functor (7.3) is an equivalence. By Proposition 7.12 the functor (7.3) identifies with the induced functor

$$ \begin{align*}{\mathrm{FUN}}(\tau^*\Delta^{n},\infty{\mathrm{CAT}}) \to\end{align*} $$

$$ \begin{align*}{\mathrm{FUN}}(\tau^*\Delta^1,\infty{\mathrm{CAT}}) \times_{{\mathrm{FUN}}(\tau^*\Delta^0,\infty{\mathrm{CAT}})} \ldots \times_{{\mathrm{FUN}}(\tau^*\Delta^0,\infty{\mathrm{CAT}})} {\mathrm{FUN}}(\tau^*\Delta^1,\infty{\mathrm{CAT}}).\end{align*} $$

The latter induces on maximal subspaces the induced map

$$ \begin{align*}\mathrm{Fun}([{n}]^{\mathrm{op}},\infty\mathrm{Cat}) \to \mathrm{Fun}([1]^{\mathrm{op}},\infty\mathrm{Cat}) \times_{\mathrm{Fun}([0]^{\mathrm{op}},\infty\mathrm{Cat})} \ldots \times_{\mathrm{Fun}([0]^{\mathrm{op}},\infty\mathrm{Cat})} \mathrm{Fun}([1]^{\mathrm{op}},\infty\mathrm{Cat}),\end{align*} $$

which is an equivalence since $[{n}] \simeq [1] \coprod _{[0]} \ldots \coprod _{[0]} [1]$ in $\infty \mathrm {Cat}.$

So it remains to see that the functor (7.3) is fully faithful. The functor (7.3) induces on mapping spaces between cocartesian fibrations ${\mathcal C} \to [{n}], {\mathcal D} \to [{n}]$ the map on maximal subspaces underlying the canonical functor

$$ \begin{align*}\mathrm{Fun}_{[{n}]}({\mathcal C},{\mathcal D}) \to\end{align*} $$

$$ \begin{align*}\mathrm{Fun}_{[1]}([1] \times_{[{n}]}{\mathcal C}, [1] \times_{[{n}]} {\mathcal D}) \times_{\mathrm{Fun}_{[0]}([0] \times_{[{n}]}{\mathcal C}, [0] \times_{[{n}]} {\mathcal D})} \ldots \mathrm{Fun}_{[1]}([1] \times_{[{n}]}{\mathcal C}, [1] \times_{[{n}]} {\mathcal D}).\end{align*} $$

The latter is an equivalence by Lemma 7.7 since every cocartesian fibration is exponentiable by Corollary 3.27.

Proof. Let $\infty {\mathrm {CAT}}'$ be the $(\infty ,2)$ -category underlying the double $\infty $ -category $\infty {\mathrm {CAT}}$ . Then the space of objects of $\infty {\mathrm {CAT}}'$ is $\iota (\infty {\mathrm {CAT}}_{[0]}) \simeq \iota (\infty \mathrm {Cat})$ and for every ${\mathrm A},{\mathrm B} \in \infty \mathrm {Cat}$ there is a canonical equivalence

$$ \begin{align*}{\mathrm{Mor}}_{\infty{\mathrm{CAT}}'}({\mathrm A},{\mathrm B}) \simeq \{({\mathrm A},{\mathrm B})\} \times_{\infty{\mathrm{CAT}}^{\prime}_{[0]} \times \infty{\mathrm{CAT}}^{\prime}_{[0]}} \infty{\mathrm{CAT}}^{\prime}_{[1]} \simeq \{({\mathrm A},{\mathrm B})\} \times_{\infty{\mathrm{CAT}}_{[0]} \times \infty{\mathrm{CAT}}_{[0]}} \infty{\mathrm{CAT}}_{[1]}.\end{align*} $$

By the dual of Corollary 5.3 there is a canonical equivalence