Lemma 4.5.5.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $B$ be a simplicial set, and let $A \subseteq B$ be a simplicial subset with the property that the inclusion $A \hookrightarrow B$ is a categorical equivalence. Then every diagram $f_0: A \rightarrow \operatorname{\mathcal{C}}$ can be extended to a diagram $f: B \rightarrow \operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Corollary 4.4.5.4, the restriction map $\theta : \operatorname{Fun}(B,\operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}(A, \operatorname{\mathcal{C}})^{\simeq }$ is a Kan fibration. Since the inclusion $A \hookrightarrow B$ is a categorical equivalence, the map $\theta $ is a homotopy equivalence of Kan complexes (Proposition 4.5.3.8). Invoking Proposition 3.3.7.6, we conclude that $\theta $ is a trivial Kan fibration. In particular, it is surjective on vertices. $\square$