Variant 8.3.2.6. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories, and let $\kappa $ be an uncountable cardinal. Then the construction of Example 8.3.2.4 induces a monomorphism
\[ \xymatrix@C =50pt@R=50pt{ \{ \textnormal{Profunctors $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$} \} / \textnormal{Isomorphism} \ar [d] \\ \{ \textnormal{Couplings $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$} \} / \textnormal{Equivalence}, } \]
whose image consists of equivalence classes of couplings $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ whose fibers are essentially $\kappa $-small.