Kerodon

Corollary 5.1.2.2. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories, and let $\operatorname{N}_{\bullet }(q): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ be the induced morphism of simplicial sets. Let $g: Y \rightarrow Z$ be a morphism in the category $\operatorname{\mathcal{C}}$. Then $g$ is $q$-cartesian (in the sense of Definition 5.0.0.1) if and only if it is $\operatorname{N}_{\bullet }(q)$-cartesian (when regarded as an edge of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$).