Remark 5.3.2.3. Let $T: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ be a functor between categories, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets indexed by $\operatorname{\mathcal{C}}$, and let $\mathscr {F}'$ denote the composition $\mathscr {F} \circ T$. Then we have a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}' ) \ar [r] \ar [d]^{U'} & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [d]^{U} \\ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}') \ar [r]^{ \operatorname{N}_{\bullet }(T) } & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), } \]
where $U$ and $U'$ denote the projection maps of Construction 5.3.2.1. In particular, for every object $C \in \operatorname{\mathcal{C}}$, we have a canonical isomorphism of simplicial sets
\[ \mathscr {F}(C) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ). \]