Theorem 9.4.8.4. 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 9.4.8.3.
- $(2)$
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 $\operatorname{\mathcal{E}}^{\operatorname{op}}$.
Proof of Theorem 9.4.8.4.
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 9.4.8.8 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 9.4.8.2). By virtue of Proposition 9.4.1.16, 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.
$\square$