Remark 5.3.2.7. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets indexed by $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$ which does not belong to $\operatorname{\mathcal{C}}_0$, the simplicial set $\mathscr {F}(C)$ is empty. Then the image of the projection map $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is contained in $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 )$. Setting $\mathscr {F}_0 = \mathscr {F}|_{\operatorname{\mathcal{C}}_0}$, we deduce that the canonical map
is an isomorphism.