Kerodon

Remark 11.10.3.2. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories and let $U^{\operatorname{op}}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be the induced functor of opposite categories. Let $f: X \rightarrow Y$ be a morphism in the category $\operatorname{\mathcal{D}}$, which we identify with a morphism $f^{\operatorname{op}}: Y \rightarrow X$ in the opposite category $\operatorname{\mathcal{D}}^{\operatorname{op}}$. Then $f$ is locally $U$-cartesian if and only if $f^{\operatorname{op}}$ is locally $U^{\operatorname{op}}$-cocartesian.