Kerodon

Definition 8.3.2.7. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a profunctor. Let $X$ be an object of $\operatorname{\mathcal{C}}_{-}$, let $Y$ be an object of $\operatorname{\mathcal{C}}_{+}$, and let $\eta $ be a vertex of the Kan complex $\mathscr {K}(X,Y)$. We will say that $\eta $ is universal if it exhibits the functor $\mathscr {K}(-,Y): \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ as represented by the object $X \in \operatorname{\mathcal{C}}_{-}$. We say that $\eta $ is couniversal if it exhibits the functor $\mathscr {K}(X, -): \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ as corepresented by the object $Y \in \operatorname{\mathcal{C}}_{+}$.