# Kerodon

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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). $\square$