Kerodon

Variant 5.2.8.6 (Parametrized Contravariant Transport). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets and let $C$ and $D$ be vertices of $\operatorname{\mathcal{C}}$. Applying Proposition 5.2.8.4 to the opposite cocartesian fibration $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$, we obtain a diagram $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{D}, \operatorname{\mathcal{E}}_{C} )$, carrying each edge $f: C \rightarrow D$ to a functor $f^{\ast }: \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$ given by contravariant transport along $f$.