Kerodon

Remark 5.3.3.10 (Functoriality in $\mathscr {F}$). Let $\operatorname{\mathcal{C}}$ be a category. Then the construction $\mathscr {F} \mapsto \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ determines a functor from the diagram category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ to the category $( \operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }$ of simplicial sets over the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. This functor commutes with all limits and with filtered colimits.