Remark 4.5.6.5. Let $\operatorname{\mathcal{C}}$ be a small category and $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be an isofibrant diagram. Then, for each object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is an $\infty $-category. That is, for $0 < i < n$, every morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow \mathscr {F}(C)$ can be extended to an $n$-simplex of $\mathscr {F}(C)$. This follows by applying condition $(\ast )$ of Definition 4.5.6.3 to the functor
\[ \mathscr {E}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}\quad \quad \mathscr {E}(D) = \Delta ^ n \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D), \]
together with the subfunctor $\mathscr {E}_0 \subseteq \mathscr {E}$ given by $\mathscr {E}_0(D) = \Lambda ^{n}_{i} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$.