Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 8.4.5.6. 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:

$(0)$

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

$(1)$

The functor $f$ is fully faithful.

$(2)$

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

$(3)$

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

Proof of Proposition 8.4.5.6. By virtue of Variant 8.4.5.8, 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 5.4.8.10. $\square$