Definition 8.6.4.12. 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. We say that a morphism of simplicial sets $\mathscr {K}: \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}$ exhibits $U^{\vee }$ as a cocartesian dual of $U$ if the following conditions are satisfied:
- $(a)$
For each vertex $C \in \operatorname{\mathcal{C}}$, the induced map $\mathscr {K}_{C}: \operatorname{\mathcal{E}}^{\vee }_{C} \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{S}}$ is a balanced profunctor (see Definition 8.3.2.18).
- $(b)$
Let $f: X \rightarrow Y$ be a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$ and let $f^{\vee }: X^{\vee } \rightarrow Y^{\vee }$ be a $U^{\vee }$-cocartesian edge of $\operatorname{\mathcal{E}}^{\vee }$ having the same image $e: C \rightarrow D$ in $\operatorname{\mathcal{C}}$. Then the map of Kan complexes
\[ \mathscr {K}( f^{\vee }, f): \mathscr {K}_{C}( X^{\vee }, X) \rightarrow \mathscr {K}_{D}( Y^{\vee }, Y ) \]carries universal vertices of $\mathscr {K}_{C}(X^{\vee }, X))$ to universal vertices of $\mathscr {K}_{D}(Y^{\vee }, Y)$ (see Definition 8.3.2.7).