Kerodon

Example 11.10.3.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories and let $\operatorname{N}_{\bullet }(U): \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be the induced morphism of simplicial sets. Let $f: X \rightarrow Y$ be a morphism in the category $\operatorname{\mathcal{E}}$. It follows from Corollary 5.1.2.2 that $f$ is locally $U$-cartesian (in the sense of Definition 11.10.3.1) if and only if it is locally $\operatorname{N}_{\bullet }(U)$-cartesian when viewed as an edge of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{E}})$ (in the sense of Definition 5.1.3.1). Similarly, $f$ is locally $U$-cocartesian if and only if it is locally $\operatorname{N}_{\bullet }(U)$-cocartesian.