Definition 8.2.4.1. Let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories and let $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ be a functor. We will say that a morphism of couplings
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\begin{equation} \begin{gathered}\label{equation:representable-coupling-witness-revised} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \ar [r]^-{ \widetilde{G} } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{ \lambda } \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_{+} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ G^{\operatorname{op}} \times \operatorname{id}} & \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} } \end{gathered} \end{equation}
exhibits $\lambda $ as represented by $G$ if, for every object $C \in \operatorname{\mathcal{C}}_{+}$, the image $\widetilde{G}( \operatorname{id}_{C} )$ is a universal object of $\operatorname{\mathcal{C}}$.