Corollary 4.5.3.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $f,f': K \rightarrow \operatorname{\mathcal{C}}$ be diagrams which are isomorphic (when viewed as objects of the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$). Then $f$ is a categorical equivalence if and only if $f'$ is a categorical equivalence.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$