Remark 5.6.4.6. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}$ be a functor from $\operatorname{\mathcal{C}}$ to the category of simplicial sets. Then the diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \ar [rr]^-{\theta } \ar [dr] & & \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}) \ar [dl] \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) & } \]
is commutative, where the vertical morphisms are the projection maps of Definitions 5.3.3.1 and 5.6.2.1 and $\theta $ is the comparison morphism of Construction 5.6.4.1