Kerodon

Example 8.6.4.2. Let $\operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{E}}^{\vee }$ be $\infty $-categories, and let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^0$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \Delta ^0$ denote the projection maps. Then a left fibration $\widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}^{\vee } \times _{ \Delta ^0 } \operatorname{\mathcal{E}}= \operatorname{\mathcal{E}}^{\vee } \times \operatorname{\mathcal{E}}$ exhibits $U^{\vee }$ as a cocartesian dual of $U$ if and only if it is a balanced coupling. In particular, $U^{\vee }$ is a cocartesian dual of $U$ if and only if $\operatorname{\mathcal{E}}^{\vee }$ is equivalent to the opposite $\infty $-category $\operatorname{\mathcal{E}}^{\operatorname{op}}$ (Corollary 8.2.6.6).