Proposition 8.2.3.7 (044Z)

Proposition 8.2.3.7. Suppose we are given a morphism of couplings

8.37

\begin{equation} \begin{gathered}\label{equation:duality-functor-left-to-right2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{ \widetilde{G} } \ar [d]^{\lambda } & \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}) \ar [d]^{\mu } \\ \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} \end{equation}

The following conditions are equivalent:

$(1)$: The diagram (8.37) exhibits the coupling $\lambda $ as represented by the functor $G$ (in the sense of Definition 8.2.3.1).
$(2)$: For every object $C \in \operatorname{\mathcal{C}}_{+}$, the functor $\widetilde{G}$ induces an equivalence of $\infty $-categories

\[ \widetilde{G}_{C}: \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}_{+} } \{ C \} \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}) \times _{ \operatorname{\mathcal{C}}_{-} } \{ G(C) \} . \]
$(3)$: The coupling $\lambda $ is representable and, for every universal object $C \in \operatorname{\mathcal{C}}$, the image $\widetilde{G}(C) \in \operatorname{Tw}(\operatorname{\mathcal{C}}_{-} )$ is an isomorphism (when viewed as a morphism of the $\infty $-category $\operatorname{\mathcal{C}}_{-}$).
$(4)$: The coupling $\lambda $ is representable and the triple $(\operatorname{id}, \widetilde{G}, G)$ is initial when viewed as an object of the $\infty $-category $\{ \operatorname{id}\} \times _{ \operatorname{Fun}( \operatorname{\mathcal{C}}_{-}, \operatorname{\mathcal{C}}_{-} )^{\operatorname{op}} } \operatorname{Fun}_{\pm }( \operatorname{\mathcal{C}}, \operatorname{Tw}(\operatorname{\mathcal{C}}_{-} ) )$.

Proof. The implication $(1) \Rightarrow (2)$ is immediate. Note that, if condition $(2)$ is satisfied, then the coupling $\lambda $ is representable; the implications $(2) \Rightarrow (3) \Rightarrow (1)$ then follow from Lemma 8.2.3.6 (using the characterization of universal objects of $\operatorname{Tw}(\operatorname{\mathcal{C}}_{-} )$ given by Example 8.2.1.5). The equivalence $(3) \Leftrightarrow (4)$ follows from Theorem 8.2.2.11. $\square$