Definition 8.3.4.12. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories, let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$, and let $\mathscr {K}|_{ \operatorname{Tw}(\operatorname{\mathcal{D}})}$ denote the composite functor
Suppose we are given a natural transformation $\beta : \underline{ \Delta ^0 }_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) } \rightarrow \mathscr {K}|_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) }$, where $\underline{\Delta ^0}_{ \operatorname{Tw}(\operatorname{\mathcal{D}})}$ denotes the constant functor $\operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{S}}$ taking the value $\Delta ^0$. We say that $\beta $ exhibits the profunctor $\mathscr {K}$ as represented by $G$ if, for every object $D \in \operatorname{\mathcal{D}}$, the evaluation of $\beta $ at the object $\operatorname{id}_{D} \in \operatorname{Tw}( \operatorname{\mathcal{D}})$ determines a vertex $\beta ( \operatorname{id}_{D} ) \in \mathscr {K}( G(D), D )$ which exhibits the functor $\mathscr {K}( -, D)$ as represented by the object $G(D) \in \operatorname{\mathcal{C}}$ (see Variant 5.6.6.2).