Variant 11.5.0.40. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor and let $\sigma $ be an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, corresponding to a diagram
\[ C_0 \rightarrow C_1 \rightarrow C_2 \rightarrow \cdots \rightarrow C_ n \]
in the category $\operatorname{\mathcal{C}}$. Using Remark 5.3.3.7 and Example 5.3.3.12, we obtain an isomorphism of simplicial sets
\[ \Delta ^ n \times _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \simeq ((( \mathscr {F}(C_0) \star _{ \mathscr {F}(C_1)} \mathscr {F}(C_1)) \star _{ \mathscr {F}(C_2)} \mathscr {F}(C_2)) \star \cdots ) \star _{\mathscr {F}(C_ n)} \mathscr {F}(C_ n). \]