Exercise 8.2.2.7. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$ and $\mu : \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{D}}_{+}$ be couplings of $\infty $-categories, and suppose we are given a morphism of couplings
Show that the following conditions are equivalent:
The functors $F_{-}$, $F$, and $F_{+}$ are equivalences of $\infty $-categories.
There exists a morphism of couplings $(G_{-}, G, G_{+}) \in \operatorname{Fun}_{\pm }( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ which is a homotopy inverse to $(F_{-}, F, F_{+} )$, in the sense that the compositions $( G_{-} \circ F_{-}, G \circ F, G_{+} \circ F_{+} )$ and $(F_{-} \circ G_{-}, F \circ G, F_{+} \circ G_{+} )$ are isomorphic to $( \operatorname{id}_{\operatorname{\mathcal{C}}_{-}}, \operatorname{id}_{\operatorname{\mathcal{C}}}, \operatorname{id}_{\operatorname{\mathcal{C}}_{+}})$ and $( \operatorname{id}_{ \operatorname{\mathcal{D}}_{-}}, \operatorname{id}_{\operatorname{\mathcal{D}}}, \operatorname{id}_{ \operatorname{\mathcal{D}}_{+} } )$ as objects of the $\infty $-categories $\operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ and $\operatorname{Fun}_{\pm }( \operatorname{\mathcal{D}}, \operatorname{\mathcal{D}})$, respectively.
If these conditions are satisfied, we will say that the diagram (8.30) is an equivalence of couplings.