Kerodon

Remark 5.2.2.16. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets and let $f: C \rightarrow D$ be an edge of $\operatorname{\mathcal{C}}$. Then a functor $F: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ is given by covariant transport along $f$ if and only if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{E}}_{C}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{E}}_{D}^{\operatorname{op}}$ is given by contravariant transport along $f$ with respect to the cartesian fibration $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$.