Remark 8.3.2.22. Up to equivalence, every balanced profunctor $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ can be obtained from the construction of Corollary 8.3.2.21. More precisely, let $\operatorname{Fun}^{\mathrm{corep}}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}})$ spanned by the corepresentable functors. If $\mathscr {K}$ is balanced, then it factors as a composition
\[ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \xrightarrow {\Phi \times \operatorname{id}} \operatorname{Fun}^{\mathrm{corep}}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}}) \times \operatorname{\mathcal{C}}_{+} \xrightarrow {\operatorname{ev}} \operatorname{\mathcal{S}}, \]
where $\Phi $ is an equivalence of $\infty $-categories by virtue of Corollary 8.3.2.20.