$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 8.2.6.7. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories. Then $\lambda $ is balanced if and only if there exists an equivalence of couplings
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{\lambda } \ar [r]^-{F} & \operatorname{Tw}(\operatorname{\mathcal{D}}) \ar [d] \\ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ F_{-}^{\operatorname{op}} \times F_{+} } & \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}. } \]
Proof.
Suppose that $\lambda $ is balanced. By virtue of Theorem 8.2.6.5, the coupling $\lambda $ is corepresentable by a functor $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ which is an equivalence of $\infty $-categories. We can therefore choose a categorical pullback square
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{\lambda } \ar [r]^-{F} & \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}) \ar [d] \\ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \ar [r]^-{ F_{-}^{\operatorname{op}} \times \operatorname{id}} & \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} } \]
which exhibits $\lambda $ as corepresented by $F_{-}$. Since $F_{-}$ is an equivalence of $\infty $-categories, it follows that $F$ is also an equivalence of $\infty $-categories (Proposition 4.5.2.21). The reverse implication is an immediate consequence of Example 8.2.6.2 (and Remark 8.2.6.4).
$\square$