Kerodon

Example 5.6.1.8. Let $\operatorname{Cat}$ denote the category whose objects are (small) categories and whose morphisms are functors, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Cat}$ be a functor of ordinary categories. Composing with the nerve functor $\operatorname{N}_{\bullet }: \operatorname{Cat}\rightarrow \operatorname{Set_{\Delta }}$, we obtain a functor $\mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. There is a canonical isomorphism of simplicial sets $\operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}}) \simeq \operatorname{N}_{\bullet }( \int _{\operatorname{\mathcal{C}}} \mathscr {F} )$, where the left hand side denotes the weighted nerve of Definition 5.3.3.1 and $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denotes the category of elements introduced in Definition 5.6.1.1. See Exercise 5.3.3.17.