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Proposition Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is essentially small. Then $f$ is equivalent to the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ if and only if it satisfies the following conditions:


The $\infty $-category $\operatorname{\mathcal{D}}$ is locally small and admits small colimits.


The functor $f$ is fully faithful.


For each object $C \in \operatorname{\mathcal{C}}$, the image $f(C) \in \operatorname{\mathcal{D}}$ is atomic.


The collection of objects $\{ f(C) \} _{C \in \operatorname{\mathcal{C}}}$ is weakly dense in $\operatorname{\mathcal{D}}$.

Proof of Proposition By virtue of Variant, the only nontrivial point is to verify that if $\operatorname{\mathcal{C}}$ is an essentially small $\infty $-category, then the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is locally small. This follows from Corollary $\square$