Remark 5.3.3.7 (Functoriality in $\operatorname{\mathcal{C}}$). Let $\operatorname{\mathcal{C}}$ be a category equipped with a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, let $U: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ be a functor between categories, and let $\mathscr {F}': \operatorname{\mathcal{C}}' \rightarrow \operatorname{Set_{\Delta }}$ denote the composition $\mathscr {F} \circ U$. Then there is a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}}') \ar [r] \ar [d] & \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}' ) \ar [r]^-{ \operatorname{N}_{\bullet }(U) } & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}). } \]