Remark 8.3.4.13. In the situation of Definition 8.3.4.12, the natural transformation $\beta $ can be identified with a functor $\widetilde{G}$ which fits into a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{Tw}(\operatorname{\mathcal{D}}) \ar [r]^-{ \widetilde{G} } \ar [d] & \{ \Delta ^0 \} \operatorname{\vec{\times }}_{ \operatorname{\mathcal{S}}} ( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}) \ar [d]^{\lambda } \\ \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\ar [r]^-{ G^{\operatorname{op}} \times \operatorname{id}} & \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}. } \]
Moreover, the natural transformation $\beta $ exhibits $\mathscr {K}$ as represented by $G$ (in the sense of Definition 8.3.4.12) if and only if $\widetilde{G}$ exhibits the coupling $\lambda $ as represented by $G$ (in the sense of Definition 8.2.4.1).