Kerodon

Proposition 8.6.4.15. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of $\infty $-categories. Let $\kappa $ be an uncountable cardinal with the property that, for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-categories $\operatorname{\mathcal{E}}_{C}$ and $\operatorname{\mathcal{E}}^{\vee }_{C}$ are locally $\kappa $-small. Then $U^{\vee }$ is a cocartesian dual of $U$ if and only if there exists a morphism $\mathscr {K}: \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa }$ which exhibits $U^{\vee }$ as a cocartesian dual of $U$, in the sense of Definition 8.6.4.12.