Corollary 8.4.3.10. Let $\kappa $ be an uncountable regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\kappa $-small, and let $h^{\operatorname{\mathcal{C}}}_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ be a covariant Yoneda embedding. Suppose we are given a functor
\[ T: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }, \]
where $\lambda $ is a cardinal of exponential cofinality $\geq \kappa $. Then $T$ is representable if and only if the following conditions are satisfied:
- $(1)$
For each object $C \in \operatorname{\mathcal{C}}$, the Kan complex $T( h^{\operatorname{\mathcal{C}}}_{C} )$ is essentially $\kappa $-small.
- $(2)$
The functor $T$ preserves $\kappa $-small limits.
Moreover, if these conditions are satisfied, then the functor $T$ is representable by the object $\mathscr {F} = T \circ (h_{\bullet }^{\operatorname{\mathcal{C}}})^{\operatorname{op}}$.
Proof.
Assume first that $T$ is representable by an object $\mathscr {G} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. In this case, Corollary 7.4.1.19 guarantees that $T$ satisfies condition $(2)$. To verify $(1)$, we note that for each object $C \in \operatorname{\mathcal{C}}$, Proposition 8.3.1.3 supplies a homotopy equivalence
\[ T( h^{\operatorname{\mathcal{C}}}_{C} ) \simeq \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) }( h^{\operatorname{\mathcal{C}}}_{C}, \mathscr {G} ) \xrightarrow {\sim } \mathscr {G}(C); \]
the desired result now follows from the observation that $\mathscr {G}(C)$ is essentially $\kappa $-small.
We now prove the converse. Assume that $T$ satisfies conditions $(1)$ and $(2)$ and set $\mathscr {F} = T \circ (h_{\bullet }^{\operatorname{\mathcal{C}}})^{\operatorname{op}}$. Condition $(1)$ guarantees that we can view $\mathscr {F}$ as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. Since $\lambda $ has exponential cofinality $\geq \kappa $, the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is locally $\lambda $-small (Corollary 4.7.8.8). We can therefore choose a functor $T': \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ which is representable by $\mathscr {F}$ (Theorem 5.6.6.13). It follows from Proposition 8.4.2.5 that the composition $T' \circ (h_{\bullet }^{\operatorname{\mathcal{C}}})^{\operatorname{op}}$ is isomorphic to the functor $\mathscr {F} = T \circ (h_{\bullet }^{\operatorname{\mathcal{C}}})^{\operatorname{op}}$. Using condition $(2)$ (and Corollary 7.4.1.19), we see that the functors $T$ and $T'$ both preserve $\kappa $-small limits. Applying Theorem 8.4.3.3, we conclude that $T'$ is isomorphic to $T$, so that the functor $T$ is also representable by the object $\mathscr {F}$.
$\square$