Corollary 8.3.4.10. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories. Let $\kappa $ be an uncountable cardinal for which $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ are locally $\kappa $-small, and let $\operatorname{\mathcal{E}}\subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}}^{< \kappa } )$ denote the full subcategory spanned by those profunctors $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ which are both representable and corepresentable. Then:
- $(1)$
Composition with the covariant Yoneda embedding $\operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{ < \kappa } )$ induces an equivalence of $\infty $-categories $\rho : \operatorname{RFun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{C}}_{-} ) \rightarrow \operatorname{\mathcal{E}}$.
- $(2)$
Composition with the contravariant Yoneda embedding $\operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}}^{< \kappa } )$ induces an equivalence of $\infty $-categories $\lambda : \operatorname{LFun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{C}}_{+} ) \rightarrow \operatorname{\mathcal{E}}^{\operatorname{op}}$.
- $(3)$
The composition $[\rho ]^{-1} \circ [\lambda ^{\operatorname{op}}]$ determines a canonical isomorphism $\operatorname{LFun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{C}}_{+} )^{\operatorname{op}} \simeq \operatorname{RFun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{C}}_{-} )$ in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$, which carries each functor $F \in \operatorname{LFun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{C}}_{+} )$ to a functor $G \in \operatorname{RFun}( \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{C}}_{-} )$ which is right adjoint to $F$.