# Kerodon

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### 9.3.8 Fiberwise Cocompletion via the Yoneda Embedding

Recall that, if $\operatorname{\mathcal{E}}$ is an essentially small $\infty$-category, then the contravariant Yoneda embedding $h^{\bullet }: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{S}})$ exhibits the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{S}})$ as a cocompletion of $\operatorname{\mathcal{E}}$ (Theorem 8.4.0.3). Our goal in this section is to prove a relative version of this statement.

Proposition 9.3.8.1. Let $\kappa$ be an uncountable regular cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of simplicial sets which is essentially $\kappa$-small. Then:

$(1)$

The projection map $V: \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration of simplicial sets (see Construction 4.5.9.1).

$(2)$

Let $V^{\dagger }: \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian conjugate of $V$ (Definition 8.6.1.1). Then $V^{\dagger }$ is a fiberwise $\kappa$-cocompletion of $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$.

The first assertion of Proposition 9.3.8.1 is a special case of the following:

Lemma 9.3.8.2. Let $\kappa$ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets which is essentially $\kappa$-small, and let $\operatorname{\mathcal{D}}$ be an $\infty$-category which is $\kappa$-cocomplete. Then the projection map $V: \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration. Moreover, for every edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the covariant transport functor

$\operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{D}}) = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \{ C' \} \times _{\operatorname{\mathcal{C}}} \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) = \operatorname{Fun}( \operatorname{\mathcal{E}}_{C'}, \operatorname{\mathcal{D}})$

preserves $\kappa$-small colimits.

Proof. To show that $V$ is a cocartesian fibration, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex (Proposition 5.1.4.7). In this case, $U$ is exponentiable (Corollary 9.3.6.19), so the desired result follows from Variant 8.6.5.11. To prove the second assertion, we may assume that $\operatorname{\mathcal{C}}= \Delta ^1$ with $C = 0$ and $C' = 1$. In this case, $\operatorname{\mathcal{E}}_{C}$ and $\operatorname{\mathcal{E}}_{C'}$ are full subcategories of $\operatorname{\mathcal{E}}$, and determine restriction functors $R: \operatorname{Fun}(\operatorname{\mathcal{E}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{D}})$ and $R': \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{C'}, \operatorname{\mathcal{D}})$. The covariant transport functor $e_{!}$ can then be identified with the composition $R' \circ R^{L}$, where $R^{L}$ is left adjoint to $R$ (given by left Kan extension along the inclusion $\operatorname{\mathcal{E}}_{C} \hookrightarrow \operatorname{\mathcal{E}}$). Since the functor $R^{L}$ preserves all colimits (Corollary 7.1.3.21), we are reduced to showing that the functor $R'$ preserves $\kappa$-small colimits, which follows from the criterion of Proposition 7.1.6.1. $\square$

To prove Proposition 9.3.8.1, we may assume without loss of generality that $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is a flat inner fibration of $\infty$-categories (Corollary 9.3.6.26). In this case, the results of ยง8.6.2 provide an explicit example of a cartesian dual of the inner fibration $V$.

Construction 9.3.8.3. Let $\kappa$ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of $\infty$-categories which is essentially $\kappa$-small, and let $V: \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{C}}$ be the cocartesian fibration of Lemma 9.3.8.2. We let $\operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger }$ denote the full subcategory

$\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \subseteq \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ))$

introduced in Construction 8.6.2.2, and let $V^{\dagger }: \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ denote the projection map. It follows from Proposition 8.6.2.3 that the evaluation map

$\operatorname{ev}: \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$

exhibits $V^{\dagger }$ as a cartesian conjugate of $V$.

By virtue of Lemma 9.3.8.2 (and Corollary 9.3.6.26), Proposition 9.3.8.1 is a consequence of the following more precise assertion:

Theorem 9.3.8.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of $\infty$-categories, let $\kappa$ be an uncountable regular cardinal for which $U$ is essentially $\kappa$-small, and let

$H: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$

be a relative $\operatorname{Hom}$-functor for $U$ (see Definition 8.6.6.8), so that $H$ is classified by a map

$h: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \simeq \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) ).$

Then:

$(1)$

The functor $h$ factors through the full subcategory

$\operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger } \subseteq \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) )$

of Construction 9.3.8.3.

$(2)$

The diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}^{\operatorname{op}} \ar [rr]^-{ h } \ar [dr]_{ U^{\operatorname{op}} } & & \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger } \ar [dl]^{V^{\dagger } } \\ & \operatorname{\mathcal{C}}^{\operatorname{op}} & }$

exhibits $\operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger }$ as a fiberwise $\kappa$-cocompletion of $\operatorname{\mathcal{E}}^{\operatorname{op}}$.

We will carry out the proof of Theorem 9.3.8.4 in several steps.

Remark 9.3.8.5. Let $\kappa$ be an uncountable regular cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of $\infty$-categories which is essentially $\kappa$-small. Suppose we are given a commutative diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}^{\operatorname{op}} \ar [dr] \ar [rr]^-{h} & & \operatorname{Fun}_{/ \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \ar [dl] \\ & \operatorname{\mathcal{C}}^{\operatorname{op}}, & }$

which we identify with a morphism $H: \operatorname{\mathcal{D}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$. Using the criterion of Variant 8.6.5.11, we see that $h$ factors through $\operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger }$ if and only if $H$ satisfies the following condition:

$(\ast )$

Let $X$ be a vertex of $\operatorname{\mathcal{D}}$ having image $C \in \operatorname{\mathcal{C}}$, let $e$ be an edge of $\{ C\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$, and let $H_{e}$ denote the composition

$\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\xrightarrow {e \times \operatorname{id}} \{ C\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{D}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\xrightarrow {H} \operatorname{\mathcal{S}}^{< \kappa }.$

Then the functor $H_{e}$ is left Kan extended from $\{ 0\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.

Moreover, it suffice to verify condition $(\ast )$ under the additional assumption that $e$ belongs to the image of the equivalence $\operatorname{\mathcal{C}}_{C/} \rightarrow \{ C\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ (see Proposition 8.1.2.9).

We now prove the first assertion of Theorem 9.3.8.4:

Lemma 9.3.8.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of $\infty$-categories, let $\kappa$ be an uncountable regular cardinal for which $U$ is essentially $\kappa$-small, and let $H: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a relative $\operatorname{Hom}$-functor for $U$. Then $H$ is classified by a functor $h: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger }$.

Proof. Fix an object $X \in \operatorname{\mathcal{E}}$ having image $C = U(X)$ and an edge $e$ of the $\infty$-category $\operatorname{\mathcal{C}}_{C/}$, which we identify with a $2$-simplex $\sigma :$

$\xymatrix@R =50pt@C=50pt{ & C' \ar [dr] & \\ C \ar [ur] \ar [rr] & & C'' }$

of $\operatorname{\mathcal{C}}$, and let $H_{e}: \operatorname{N}_{\bullet }( \{ C' < C'' \} ) \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be as in condition $(\ast )$ of Remark 9.3.8.5; we wish to show that the functor $F_{e}$ is left Kan extended from the subcategory $\{ C' \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}= \Delta ^2$ and that $\sigma$ is the identity morphism. In this case, we can identify $F_{e}$ with the composition

$\operatorname{N}_{\bullet }( \{ 1 < 2 \} ) \times _{ \Delta ^2 } \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{E}}\xrightarrow {\mathscr {F}} \operatorname{\mathcal{S}}^{< \kappa },$

where $\mathscr {F}$ is corepresented by the object $X \in \operatorname{\mathcal{E}}$. Since $U$ is flat, the desired result is a reformulation of Lemma 9.3.7.2. $\square$

We now prove a special case of Theorem 9.3.8.4.

Lemma 9.3.8.7. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of $\infty$-categories, let $\kappa$ be an uncountable regular cardinal for which $U$ is essentially $\kappa$-small, and let

$H: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$

be a relative $\operatorname{Hom}$-functor for $U$. Then the induced map $h: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger }$ exhibits $\operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ as a fiberwise $\kappa$-cocompletion of $\operatorname{\mathcal{E}}^{\operatorname{op}}$ (relative to $\operatorname{\mathcal{C}}^{\operatorname{op}}$).

Proof. Let us identify $H$ with a functor $F: \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$. If $X$ an object of $\operatorname{\mathcal{E}}$ having image $C = U(X)$ and $u: C \rightarrow D$ is a morphism of $\operatorname{\mathcal{C}}$, our assumption that $U$ is a cocartesian fibration guarantees that we can lift $u$ to a $U$-cocartesian morphism $\widetilde{u}: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$. In this case, we can identify $F(X,u)$ with a functor $\operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ which is corepresented by the object $Y$. It follows that $F$ factors through the full subcategory $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \subseteq \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ of Construction 8.6.5.6. Unwinding the definitions, we have a commutative diagram

9.46
$$\begin{gathered}\label{equation:relative-Yoneda-determines} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{F} \ar [d]^{h \times \operatorname{id}} & \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \ar [d] \\ \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{\operatorname{ev}} & \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ). } \end{gathered}$$

Let $V: \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{C}}$ and $V^{\dagger }: \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ denote the projection maps, and set $V^{\operatorname{corep}} = V|_{ \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) }$. Then the lower horizontal map exhibits $V^{\dagger }$ as a cartesian conjugate of $V$ (Proposition 8.6.2.3), and the upper horizontal map exhibits $U^{\operatorname{op}}$ as a cartesian conjugate of $V^{\operatorname{corep}}$ (Theorem 8.6.6.15). Moreover, the inclusion map $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \subseteq \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ exhibits $\operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ as a fiberwise $\kappa$-cocompletion of $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ (Example 9.3.2.5). Applying Proposition 9.3.3.3, we conclude that $h$ exhibits $\operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger }$ as a fiberwise $\kappa$-cocompletion of $\operatorname{\mathcal{E}}^{\operatorname{op}}$. $\square$

To prove Theorem 9.3.8.4, we need the following relative version of Yoneda's lemma:

Lemma 9.3.8.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of $\infty$-categories which is essentially $\kappa$-small. Then the functor $h: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })^{\dagger }$ of Lemma 9.3.8.6 is fully faithful.

Proof. Using Theorem 9.3.4.1, we can choose a diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}^{\operatorname{op}} \ar [dr]_{U^{\operatorname{op}}} \ar [rr]^-{F^{\operatorname{op}}} & & \widehat{\operatorname{\mathcal{E}}}^{\operatorname{op}} \ar [dl]^{ \widehat{U}^{\operatorname{op}} } \\ & \operatorname{\mathcal{C}}^{\operatorname{op}} & }$

which exhibits $\widehat{\operatorname{\mathcal{E}}}^{\operatorname{op}}$ as a fiberwise $\kappa$-cocompletion of $\operatorname{\mathcal{E}}^{\operatorname{op}}$. Since $U$ is flat, the inner fibration $U^{\operatorname{op}}$ is also flat (Remark 9.3.6.5). Applying the criterion of Proposition 9.3.6.9, we see that $\widehat{U}^{\operatorname{op}}$ is a cartesian fibration. Let $\lambda$ be a regular cardinal of exponential cofinality $\geq \kappa$ such that $\widehat{U}$ is essentially $\lambda$-small. Let $h^{+}$ denote the composition of $h$ with the inclusion functor

\begin{eqnarray*} \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })^{\dagger } & \hookrightarrow & \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) ) \\ & \hookrightarrow & \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } ) ). \end{eqnarray*}

We will complete the proof by showing that $h^{+}$ is fully faithful.

Since $\lambda$ has exponential cofinality $\geq \kappa$, the $\infty$-category $\operatorname{\mathcal{S}}^{< \lambda }$ admits $\kappa$-small limits (Example 7.6.7.4). Let $\operatorname{Fun}( \widehat{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } )$ denote the relative exponential of Construction 4.5.9.1. By definition, objects of $\operatorname{Fun}( \widehat{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } )$ can be identified with pairs $(C, \mathscr {G} )$, where $C$ is an object of $\operatorname{\mathcal{C}}$ and $\mathscr {G}: \widehat{\operatorname{\mathcal{E}}}_{C} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ is a functor. Let $\operatorname{Fun}'( \widehat{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } )$ be the full subcategory spanned by those pairs $(C, \mathscr {F} )$, where the functor $\mathscr {G}$ preserves $\kappa$-small limits. Let $\widehat{H}: \widehat{\operatorname{\mathcal{E}}}^{\operatorname{op}} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ be a relative $\operatorname{Hom}$-functor for $\widehat{U}$, which we identify with a map

$\widehat{h}: \widehat{\operatorname{\mathcal{E}}}^{\operatorname{op}} \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Fun}( \widehat{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } ).$

Since corepresentable functors preserve limits (Corollary 7.4.5.18), the functor $\widehat{h}$ factors through the full subcategory $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Fun}'( \widehat{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } ))$. Since $F$ is fully faithful (Remark 9.3.1.14), the functor $h^{+}$ is isomorphic to the composition

\begin{eqnarray*} \operatorname{\mathcal{E}}^{\operatorname{op}} & \xrightarrow { F^{\operatorname{op}} } & \widehat{\operatorname{\mathcal{E}}}^{\operatorname{op}} \\ & \xrightarrow { \widehat{h} } & \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Fun}'( \widehat{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } )) \\ & \xrightarrow { T } & \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } )), \end{eqnarray*}

where $T$ is given by precomposition with $F$. It follows from the universal property of Theorem 9.3.1.20 that precomposition with $F$ induces an equivalence $\operatorname{Fun}'( \widehat{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } )$ of inner fibrations over $\operatorname{\mathcal{C}}$, so that $T$ is an equivalence of $\infty$-categories. To complete the proof, it will suffice to show that the functor $\widehat{h}$ is fully faithful. In other words, we can replace $U$ by $\widehat{U}$ (and $\kappa$ by the cardinal $\lambda$) and thereby reduce to proving Lemma 9.3.8.8 under the assumption that $U$ is a cocartesian fibration. In this case, the desired result follows from Lemma 9.3.8.7 (and Remark 9.3.1.14). $\square$

Proof of Theorem 9.3.8.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of $\infty$-categories which is essentially $\kappa$-small. It follows from Lemma 9.3.8.8 that a relative $\operatorname{Hom}$-functor for $U$ determines a fully faithful functor $h: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })^{\dagger }$. We wish to show that the diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}^{\operatorname{op}} \ar [rr]^-{h} \ar [dr]_{ U^{\operatorname{op}} } & & \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })^{\dagger } \ar [dl]^{V^{\dagger } } \\ & \operatorname{\mathcal{C}}^{\operatorname{op}} & }$

exhibits $\operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger }$ as a fiberwise $\kappa$-cocompletion of $\operatorname{\mathcal{E}}^{\operatorname{op}}$. By construction, $V^{\dagger }$ is a cartesian fibration which is conjugate to $V: \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{C}}$. In particular, for every morphism $f: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the contravariant transport functor

$\{ C \} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })^{\dagger } \rightarrow \{ C' \} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })^{\dagger }$

for the cartesian fibration $V^{\dagger }$ can be identified with the covariant transport functor

$\{ C \} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa }) \rightarrow \{ C' \} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })$

for the cocartesian fibration $V$ (see Proposition 8.6.1.5), and therefore preserves $\kappa$-small colimits (Lemma 9.3.8.2). By virtue of Proposition 9.3.1.16, it will suffice to show that for each object $C \in \operatorname{\mathcal{C}}$, the map of fibers

$h_{C}: \operatorname{\mathcal{E}}_{C}^{\operatorname{op}} \rightarrow \{ C\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger }$

exhibits $\{ C\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger }$ as a $\kappa$-cocompletion of $\operatorname{\mathcal{E}}_{C}^{\operatorname{op}}$. Unwinding the definitions, we see that the composition of $h_{C}$ with the equivalence

$\{ C\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger } \simeq \{ C\} \times _{ \operatorname{\mathcal{C}}} \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$

identifies with the contravariant Yoneda embedding for the $\infty$-category $\operatorname{\mathcal{E}}_{C}$. The desired result now follows from Theorem 8.4.3.3. $\square$