# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark 5.5.6.6. 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

$\xymatrix@R =50pt@C=50pt{ \int ^{\mathrm{s}}_{\operatorname{\mathcal{C}}}\mathscr {F} \ar [rr]^-{\theta } \ar [dr] & & \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}) \ar [dl] \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) & }$

is commutative, where the vertical morphisms are the projection maps of Definitions 5.5.3.1 and 5.5.4.1 and $\theta$ is the comparison morphism of Construction 5.5.6.1