Proposition 7.5.6.12. Let $\operatorname{\mathcal{C}}$ be a small category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, and let $\mathscr {F}_{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be the diagram of Construction 7.5.6.8. Then there is a canonical isomorphism of simplicial sets $\lambda : \varinjlim ( \mathscr {F}_{+} ) \rightarrow \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ which is characterized by the following requirement: for each object $C \in \operatorname{\mathcal{C}}$, the composition
\begin{eqnarray*} \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) & = & \mathscr {F}_{+}(C) \\ & \rightarrow & \varinjlim ( \mathscr {F}_{+} ) \\ & \xrightarrow {\lambda } & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \end{eqnarray*}
is given by projection onto the second factor.