Variant 8.2.3.8 (0451)

Variant 8.2.3.8. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories and let $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ be a functor. We say that $\lambda $ is corepresentable by $F$ if there exists a categorical pullback square

8.38

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

In this case, we will say that the diagram (8.38) exhibits the coupling $\lambda $ as corepresented by $F$. It follows from Theorem 8.2.3.4 that $\lambda $ is corepresentable (in the sense of Definition 8.2.1.3) if and only if it is corepresentable by $F$, for some functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$. Moreover, if this condition is satisfied, then the functor $F$ is uniquely determined up to isomorphism.