Kerodon

Remark 8.6.4.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, and let $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be a cocartesian dual of $U$. Then, for every morphism of simplicial sets $\operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$, the projection map $U_0^{\vee }: \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}_0$ is a cocartesian dual of the projection map $U_0: \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}_0$. In particular, for every object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}^{\vee }_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}^{\vee }$ is equivalent to the opposite of the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ (Example 8.6.4.2).