Proposition 4.5.6.19. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be an isofibrant diagram. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is a Kan complex. Then, for every diagram $\mathscr {E}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, the simplicial set $X = \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F} )_{\bullet }$ is a Kan complex.
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Proof. By virtue of Corollary 4.5.6.12, the simplicial set $X$ is an $\infty $-category. Define $\mathscr {F}^{\Delta ^{1}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ by the formula $\mathscr {F}^{ \Delta ^{1} }(C) = \operatorname{Fun}( \Delta ^1, \mathscr {F}(C) )$. Then $\mathscr {F}^{ \Delta ^1 }$ is also an isofibrant diagram. Moreover, our assumption that each $\mathscr {F}(C)$ is a Kan complex guarantees that the diagonal embedding $\mathscr {F} \hookrightarrow \mathscr {F}^{\Delta ^1}$ is a levelwise categorical equivalence. Applying Corollary 4.5.6.16, we deduce that the diagonal map $X \hookrightarrow \operatorname{Fun}( \Delta ^1, X)$ is an equivalence of $\infty $-categories. In particular, every morphism of $X$ is isomorphic (as an object of the $\infty $-category $\operatorname{Fun}( \Delta ^1, X)$ ) to an identity morphism, and is therefore an isomorphism (Example 4.4.1.14). Applying Proposition 4.4.2.1, we deduce that $X$ is a Kan complex. $\square$