Corollary 8.3.4.21 (The Universal Mapping Property of Representable Profunctors). Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories. Suppose we are given a pair of profunctors $\mathscr {K}, \mathscr {K}': \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$, and let $\beta : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{D}})} \rightarrow \mathscr {K}|_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) }$ be a natural transformation which exhibits $\mathscr {K}$ as represented by $G$. Then precomposition with $\beta $ induces a homotopy equivalence of Kan complexes
\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}) }( \mathscr {K}, \mathscr {K}') \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{Tw}( \operatorname{\mathcal{D}}), \operatorname{\mathcal{S}})}( \underline{ \Delta ^0 }_{ \operatorname{Tw}( \operatorname{\mathcal{D}})}, \mathscr {K}'|_{ \operatorname{Tw}(\operatorname{\mathcal{D}})} ). \]