Remark 8.2.2.8. 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:
The coupling $\lambda $ is representable if and only if the coupling $\mu $ is representable.
The coupling $\lambda $ is corepresentable if and only if the coupling $\mu $ is corepresentable.
An object $C \in \operatorname{\mathcal{C}}$ is universal (with respect to the coupling $\lambda $) if and only if $F(C)$ is universal $\operatorname{\mathcal{D}}$ (with respect to the coupling $\mu $).
An object $C \in \operatorname{\mathcal{C}}$ is universal (with respect to the coupling $\lambda $) if and only if $F(C)$ is universal $\operatorname{\mathcal{D}}$ (with respect to the coupling $\mu $).