Kerodon

Remark 8.2.4.7. Stated more informally, Theorem 8.2.4.6 states that if a coupling $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ is corepresented by a functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ and a coupling $\mu : \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{+}$ is represented by a functor $G: \operatorname{\mathcal{D}}_{+} \rightarrow \operatorname{\mathcal{D}}_{-}$, then the objects of $\operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ can be identified with triples $(T_{-}, T_{+}, \alpha )$ where $T_{+}$ is a functor from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{D}}_{+}$, and $\alpha : T_{-} \rightarrow G \circ T_{+} \circ F$ is a natural transformation of functors from $\operatorname{\mathcal{C}}_{-}$ to $\operatorname{\mathcal{D}}_{-}$. Moreover, the natural transformation $\alpha $ is an isomorphism if and only if the corresponding functor $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ carries couniversal objects to universal objects.