Proposition 8.3.4.1 (03NR): Classification of Representable Profunctors

Proposition 8.3.4.1 (Classification of Representable Profunctors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. Let $\kappa $ be an uncountable cardinal for which $\operatorname{\mathcal{C}}$ is locally $\kappa $-small, and let

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( -, -): \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa } \]

be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ (see Notation 8.3.3.7). Then the construction $G \mapsto \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( -, G(-) )$ determines a fully faithful functor

\[ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}^{<\kappa } ), \]

whose essential image is spanned by the representable profunctors from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$.

Proof. Let $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ spanned by the representable functors. By virtue of Theorem 8.3.3.13, the construction $Y \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( -, Y)$ determines an equivalence of $\infty $-categories $h^{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. It follows that postcomposition with $h^{\bullet }$ induces an equivalence of $\infty $-categories

\[ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})), \]

which is a restatement of Proposition 8.3.4.1. $\square$