Proposition 4.5.6.14. Let $\operatorname{\mathcal{C}}$ be a small category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be an isofibrant diagram, and let $\alpha : \mathscr {E} \rightarrow \mathscr {E}'$ be a natural transformation between diagrams $\mathscr {E}, \mathscr {E}': \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. If $\alpha $ is a levelwise categorical equivalence, then precomposition with $\alpha $ induces an equivalence of $\infty $-categories
\[ \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}', \mathscr {F})_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F})_{\bullet }. \]
Proof.
Using Exercise 3.1.7.11, we can choose a contractible Kan complex $X$ containing a pair of vertices $x,y \in X$ with $x \neq y$. Evaluation at the vertices $x$ and $y$ determine trivial Kan fibrations of $\infty $-categories
\[ \operatorname{ev}_{x}, \operatorname{ev}_{y}: \operatorname{Fun}(X, \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F})_{\bullet } ) \rightarrow \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F})_{\bullet }. \]
Form a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}\ar [r]^-{ T } \ar [d]^{U} & \operatorname{Fun}(X, \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F})_{\bullet } ) \ar [d]^{ \operatorname{ev}_{x} } \\ \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}', \mathscr {F})_{\bullet } \ar [r]^-{ \circ \alpha } & \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F})_{\bullet },} \]
so that $U$ is also a trivial Kan fibration and therefore an equivalence of $\infty $-categories. It will therefore suffice to show that $\operatorname{ev}_{x} \circ T$ is an equivalence of $\infty $-categories. Since the functors $\operatorname{ev}_{x}$ and $\operatorname{ev}_{y}$ are isomorphic, this is equivalent to the requirement that $\operatorname{ev}_{y} \circ T$ is an equivalence of $\infty $-categories. In fact, the functor $\operatorname{ev}_{y} \circ T$ is a trivial Kan fibration: this follows by applying Proposition 4.5.6.11 to the levelwise categorical equivalence
\[ \underline{ \{ y\} } \times \mathscr {E} \hookrightarrow ( \underline{X} \times \mathscr {E} ) \coprod _{ ( \underline{ \{ x\} } \times \mathscr {E} ) } \mathscr {E}'. \]
$\square$