Theorem 8.2.0.4. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories. Then the assignment $G \mapsto \lambda _{G}$ of Construction 8.2.0.3 induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$} \} / \textnormal{Isomorphism} \ar [d] \\ \{ \textnormal{Representable couplings $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$} \} / \textnormal{Equivalence}. } \]