Corollary 5.4.6.9. Let $\operatorname{\mathcal{C}}$ be a minimal $\infty $-category and let $\kappa $ be an uncountable cardinal. Then $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small if and only if it is $\kappa $-small.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Suppose that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small. Then there exists an equivalence of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ is $\kappa $-small. Since $\operatorname{\mathcal{C}}$ is minimal, the functor $F$ is a monomorphism of simplicial sets (Lemma 5.4.6.8), so that $\operatorname{\mathcal{C}}$ is also $\kappa $-small (Remark 5.4.4.8). $\square$