Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 4.7.5.13. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\kappa $-small. Then the core $\operatorname{\mathcal{C}}^{\simeq }$ is an essentially $\kappa $-small Kan complex.

Proof. Since $\operatorname{\mathcal{C}}^{\simeq }$ is a replete subcategory of $\operatorname{\mathcal{C}}$ (Proposition 4.4.3.6), this is a special case of Proposition 4.7.5.12. $\square$