Variant 8.4.4.4. Let $\kappa $ be an uncountable regular cardinal and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Assume that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small, that $\operatorname{\mathcal{D}}$ admits $\kappa $-small colimits, and that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(C), D)$ is essentially $\kappa $-small for every pair of objects $C \in \operatorname{\mathcal{C}}$, $D \in \operatorname{\mathcal{D}}$. Then the functor
\[ G: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \quad \quad D \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(\bullet ), D ) \]
admits a left adjoint $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{D}}$ which preserves $\kappa $-small colimits. Moreover, the composite functor $\operatorname{\mathcal{C}}\xrightarrow { h^{\operatorname{\mathcal{C}}}_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \xrightarrow {F} \operatorname{\mathcal{D}}$ is isomorphic to $f$.
Proof.
We first prove the existence of the functor $F$. Fix a cardinal $\lambda $ of exponential cofinality $\geq \kappa $, so that the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is locally $\lambda $-small (see Corollary 4.7.8.8). By virtue of Proposition 6.2.4.1, it will suffice to show that for every functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$, the composite functor
\[ \operatorname{\mathcal{D}}\xrightarrow {G} \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \xrightarrow { \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }) }( \mathscr {F}, \bullet ) } \operatorname{\mathcal{S}}^{< \lambda } \]
is corepresentable by an object of $\operatorname{\mathcal{D}}$. Since $\operatorname{\mathcal{D}}$ admits $\kappa $-small colimits, the collection of functors $\mathscr {F}$ which satisfy this condition is closed under $\kappa $-small colimits (Remark 8.3.3.16). Using Corollary 8.4.3.9, we can reduce to the case where the functor $\mathscr {F}$ is representable by an object $C \in \operatorname{\mathcal{C}}$. In this case, the object $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })$ corepresents the evaluation functor $\operatorname{ev}_{C}: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ (Remark 8.3.1.5). It now follows from the definition of the functor $G$ that the composition $\operatorname{ev}_{C} \circ G$ is corepresentable by the object $f(C) \in \operatorname{\mathcal{D}}$.
Choose functor $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\epsilon : (F \circ G) \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ which exhibits $F$ as a left adjoint to the functor $G$. It follows from Corollary 7.1.3.21 that the functor $F$ preserves all colimits which exist in $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$; in particular, it preserves $\kappa $-small colimits. We will complete the proof by showing that $F \circ h_{\bullet }^{\operatorname{\mathcal{C}}}$ is isomorphic to $f$.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, let $\alpha _{X,Y}$ denote the morphism of Kan complexes
\[ h_{Y}^{\operatorname{\mathcal{C}}}(X) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(X), f(Y) ) = G( f(Y) )(X). \]
By virtue of Corollary 8.3.5.8, we can promote the construction $(X,Y) \mapsto \alpha _{X,Y}$ to a natural transformation of functors $\alpha : h_{\bullet }^{\operatorname{\mathcal{C}}} \rightarrow G \circ f$. Let $\beta $ denote a composition of the natural transformations
\[ F \circ h_{\bullet }^{\operatorname{\mathcal{C}}} \xrightarrow { F(\alpha ) } F \circ G \circ f \xrightarrow {\epsilon } \operatorname{id}_{\operatorname{\mathcal{D}}} \circ f = f. \]
We claim that $\beta $ is an isomorphism in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. By virtue of Theorem 4.4.4.4, it will suffice to show that $\beta $ induces an isomorphism $\beta _{X}: F( h_{X}^{\operatorname{\mathcal{C}}} ) \rightarrow f(X)$ for each object $X \in \operatorname{\mathcal{C}}$. Fix an object $D \in \operatorname{\mathcal{D}}$; we wish to show that precomposition with $\beta _{X}$ induces a homotopy equivalence of Kan complexes
\[ \beta _{X,D}: \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(X), D) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F( h_{X}^{\operatorname{\mathcal{C}}} ),D ) \simeq \operatorname{Hom}_{\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })}( h_{X}^{\operatorname{\mathcal{C}}}, G(D) ) \]
We conclude by observing that $\beta _{X,D}$ is left homotopy inverse to the morphism
\[ \operatorname{Hom}_{\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })}( h_{X}^{\operatorname{\mathcal{C}}}, G(D) ) \rightarrow G(D)(X) \simeq \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(X), D) \]
given by evaluation at $\operatorname{id}_{X} \in h_{X}^{\operatorname{\mathcal{C}}}(X)$, which is a homotopy equivalence by virtue of Proposition 8.3.1.3
$\square$