Remark 5.3.4.12. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. Then the diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \ar [rr]^{\lambda _{t}} \ar [dr] & & \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \ar [dl] \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) & } \]
commutes, where $\lambda _{t}$ is the morphism of Construction 5.3.4.11 and the vertical morphism are the projection maps of Construction 5.3.2.1 and Definition 5.3.3.1.