Remark 8.3.3.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. By virtue of Proposition 8.3.3.2, the following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is locally small.
The $\infty $-category $\operatorname{\mathcal{C}}$ admits a covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$.
The $\infty $-category $\operatorname{\mathcal{C}}$ admits a contravariant Yoneda embedding $h^{\bullet }: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$.
If these conditions are satisfied, then the functors $h_{\bullet }$ and $h^{\bullet }$ are uniquely determined up to isomorphism. Moreover, for every object $X \in \operatorname{\mathcal{C}}$, the functor $h_{X}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is representable by $X$, and the functor $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable by $X$ (Proposition 8.3.5.5).