Kerodon

Example 5.1.4.2. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between ordinary categories. Then $q$ is a cartesian fibration (in the sense of Definition 5.0.0.3) if and only if the induced morphism of simplicial sets $\operatorname{N}_{\bullet }(q): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is a cartesian fibration (in the sense of Definition 5.1.4.1). Similarly, $q$ is a cocartesian fibration if and only if $\operatorname{N}_{\bullet }(q)$ is a cocartesian fibration of simplicial sets. See Corollary 5.1.2.2.