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 $\operatorname{\mathcal{D}}$.