# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

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.

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). $\square$