Proposition 8.3.4.20 (Representable Profunctors as Kan Extensions). 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, 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 $\beta $ exhibits $\mathscr {K}$ as a left Kan extension of the constant diagram $\underline{ \Delta ^0 }_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) }$ along the composite map
Proof. Let $\operatorname{\mathcal{E}}$ denote the oriented fiber product $\{ \Delta ^0 \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} ( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}})$ and let $\mu : \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}$ be the projection onto the second factor, so that we have a tautological natural transformation $\widetilde{\beta }: \underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {K} \circ \mu $. It follows from Proposition 7.4.2.13 that $\widetilde{\beta }$ exhibits $\mathscr {K}$ as a left Kan extension of $\underline{\Delta ^0}_{\operatorname{\mathcal{E}}}$ along $\mu $. The natural transformation $\beta $ then determines a functor $T: \operatorname{Tw}( \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{E}}$ such that precomposition with $T$ carries $\widetilde{\beta }$ to $\beta $. By the transitivity of the formation of of Kan extensions (Proposition 7.3.8.18), we are reduced to showing that the identity transformation $\operatorname{id}: \underline{ \Delta ^0}_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) } \rightarrow \underline{ \Delta ^{0} }_{\operatorname{\mathcal{E}}} \circ T$ exhibits $\underline{ \Delta ^0 }_{\operatorname{\mathcal{E}}}$ as a left Kan extension of $\underline{ \operatorname{Tw}( \operatorname{\mathcal{D}}) }$ along $T$. This is a special case of Example 7.3.1.9, since the functor $T$ is left cofinal (Proposition 8.3.4.19). $\square$