Proposition 4.7.5.12. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\kappa $-small. Then any replete subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ is also essentially $\kappa $-small.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Choose an equivalence of $\infty $-categories $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{D}}$ is $\kappa $-small. Then the inverse image $\operatorname{\mathcal{D}}_0 = F^{-1}( \operatorname{\mathcal{C}}_0 )$ is $\kappa $-small (Remark 4.7.4.8), and the functor $F$ restricts to an equivalence of $\infty $-categories $\operatorname{\mathcal{D}}_0 \rightarrow \operatorname{\mathcal{C}}_0$ (Corollary 4.5.2.29). $\square$