Kerodon

Remark 8.6.4.3 (Symmetry). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets. Then a left fibration $\lambda = \widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ exhibits $U^{\vee }$ as a cocartesian dual of $U$ if and only if it exhibits $U$ as a cocartesian dual of $U^{\vee }$, after identifying $\operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ with $\operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}^{\vee }$. In particular, $U^{\vee }$ is a cocartesian dual of $U$ if and only if $U$ is a cocartesian dual of $U^{\vee }$.