Kerodon

Example 8.6.5.17. In the situation of Definition 8.6.5.16, the morphism $\mathscr {K}$ exhibits $U^{\vee }$ as a cocartesian dual of $U$ (in the sense of Variant 8.6.4.13) if and only if it is a $\operatorname{\mathcal{C}}$-family of corepresentable profunctors having the further property that each of the profunctors $\mathscr {K}_{C}$ is balanced: that is, it is corepresentable by an equivalence of $\infty $-categories $(\operatorname{\mathcal{E}}^{\vee }_{C})^{\operatorname{op}} \rightarrow \operatorname{\mathcal{E}}_{C}$ (see Corollary 8.3.2.20).