Kerodon

Remark 11.10.1.12. Let $q: X \rightarrow S$ and $q': X' \rightarrow S$ be cartesian fibrations of simplicial sets. Then a morphism $f: X \rightarrow X'$ is an equivalence of cartesian fibrations over $S$ if and only if the opposite morphism $f^{\operatorname{op}}: X^{\operatorname{op}} \rightarrow X'^{\operatorname{op}}$ is an equivalence of cocartesian fibrations over $S^{\operatorname{op}}$. In particular, the cartesian fibrations $q$ and $q'$ are equivalent if and only if the cocartesian fibrations $q^{\operatorname{op}}: X^{\operatorname{op}} \rightarrow S^{\operatorname{op}}$ and $q'^{\operatorname{op}}: X'^{\operatorname{op}} \rightarrow S^{\operatorname{op}}$ are equivalent.