Variant 8.2.4.5 (045G)

Variant 8.2.4.5. 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 $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ be a functor. We will say that a morphism of couplings

8.46

\begin{equation} \begin{gathered}\label{equation:representable-coupling-witness-revised2} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{-} ) \ar [r]^-{ \widetilde{F} } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{ \lambda } \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{-} \ar [r]^-{\operatorname{id}\times F} & \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} } \end{gathered} \end{equation}

exhibits $\lambda $ as corepresented by $F$ if, for every object $X_{-} \in \operatorname{\mathcal{C}}_{-}$, the image $\widetilde{F}( \operatorname{id}_{X_{-}} )$ is a couniversal object of $\operatorname{\mathcal{C}}$. Equivalently, the diagram (8.46) exhibits $\lambda $ as corepresented by $F$ if exhibits the coupling $\lambda ': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{+} \times \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}$ of Remark 8.2.1.4 as represented by the functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}_{+}^{\operatorname{op}}$.