Example 8.3.4.8. Let $\kappa $ be an uncountable cardinal, let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be locally $\kappa $-small $\infty $-categories, and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a profunctor from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$. The following conditions are equivalent:
The profunctor $\mathscr {K}$ is balanced (Definition 8.3.2.18).
The profunctor $\mathscr {K}$ is representable by a functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ which is an equivalence of $\infty $-categories.
The profunctor $\mathscr {K}$ is corepresentable by a functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ which is an equivalence of $\infty $-categories.
By virtue of Theorem 8.3.3.13, this is a reformulation of Corollary 8.3.2.20.