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Definition (Balanced Profunctors). Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories. We say that a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ is balanced if it satisfies the following conditions:

  • The profunctor $\mathscr {K}$ is representable and corepresentable (Definition

  • 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 $\mathscr {K}(X,Y)$. Then $\eta $ is universal if and only if it is couniversal.