Remark 8.2.6.4. Suppose we are given a morphism of couplings
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^{\lambda } & \operatorname{\mathcal{D}}\ar [d]^{\mu } \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{F^{\operatorname{op}}_{-} \times F_{+}} & \operatorname{\mathcal{D}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{D}}_{+} } \]
which is an equivalence (in the sense of Exercise 8.2.2.7). Then $\lambda $ is balanced if and only if $\mu $ is balanced. See Remark 8.2.2.8.