Corollary 4.5.6.16. Let $\operatorname{\mathcal{C}}$ be a small category, let $\mathscr {E}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a levelwise categorical equivalence between diagrams $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. If $\mathscr {F}$ and $\mathscr {G}$ are isofibrant, then composition with $\alpha $ induces an equivalence of $\infty $-categories
\[ \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F})_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {G})_{\bullet }. \]