Variant 8.3.3.12. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{S}}^{< \kappa }$ denote the $\infty $-category of $\kappa $-small spaces (see Variant 5.5.4.13). For every $\infty $-category $\operatorname{\mathcal{C}}$, the following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is locally $\kappa $-small.
The $\infty $-category $\operatorname{\mathcal{C}}$ admits a covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$.
The $\infty $-category $\operatorname{\mathcal{C}}$ admits a contravariant Yoneda embedding $h^{\bullet }: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$.
See Variant 8.3.5.7.