$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Variant 8.4.5.8. Let $\kappa $ be an uncountable regular cardinal and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small. Then $f$ is equivalent to the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{<\kappa })$ if and only if it satisfies the following conditions:
- $(0)$
The $\infty $-category $\operatorname{\mathcal{D}}$ admits $\kappa $-small colimits.
- $(1)$
The functor $f$ is fully faithful.
- $(2)$
Let $C$ be an object of $\operatorname{\mathcal{C}}$. Then the image $f(C) \in \operatorname{\mathcal{D}}$ corepresents a functor
\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa } \quad \quad Y \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(C), Y) \]
which preserves $\kappa $-small colimits.
- $(3)$
The collection of objects $\{ f(X) \} _{X \in \operatorname{\mathcal{C}}}$ is weakly dense in $\operatorname{\mathcal{D}}$.
Proof of Variant 8.4.5.8.
Let $\kappa $ be an uncountable regular cardinal, let $\operatorname{\mathcal{C}}$ be an essentially $\kappa $-small $\infty $-category, and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which satisfies the hypotheses of Variant 8.4.5.8. By virtue of Lemma 8.4.5.10, there exists a fully faithful functor $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{D}}$ which preserves $\kappa $-small colimits such that $f$ is isomorphic to the composite functor
\[ \operatorname{\mathcal{C}}\xrightarrow { h_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \xrightarrow { F} \operatorname{\mathcal{D}}. \]
To complete the proof, it will suffice to show that $F$ is an equivalence of $\infty $-categories. Using Variant 8.4.4.2, we see that the functor $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$, given on objects by the formula $G(D)(C) = \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(C), D)$. By virtue of Corollary 6.3.3.14, it will suffice to show that the functor $G$ is conservative. This follows from our assumption that the collection of objects $\{ f(C) \} _{C \in \operatorname{\mathcal{C}}}$ is weakly dense in $\operatorname{\mathcal{D}}$ (Remark 8.4.5.4).
$\square$