# Kerodon

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Definition 8.2.3.5. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty$-categories and let $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ be a functor. We say that a morphism of couplings

8.21
$$\begin{gathered}\label{equation:duality-functor-right-to-left} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{ \widetilde{G} } \ar [d]^{\lambda } & \operatorname{Tw}( \operatorname{\mathcal{C}}_{-} ) \ar [d] \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ \operatorname{id}\times G } & \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{-}, } \end{gathered}$$

exhibits the coupling $\lambda$ as represented by $G$ if it is a categorical pullback square. Note that $\lambda$ is representable by $G$ if and only if there exists a morphism of couplings which exhibits $\lambda$ as represented by $G$.