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Lemma 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 $f: B \rightarrow \operatorname{\mathcal{C}}$.

Proof. By virtue of Corollary, 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 Invoking Proposition, we conclude that $\theta $ is a trivial Kan fibration. In particular, it is surjective on vertices. $\square$