Example 5.3.3.14. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. For every morphism $f: C \rightarrow D$ in $\operatorname{\mathcal{C}}$, Remark 5.3.3.7 and Example 5.3.3.13 supply an isomorphism of simplicial sets
\[ \Delta ^1 \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \simeq \mathscr {F}(C) \star _{ \mathscr {F}(D) } \mathscr {F}(D). \]