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Theorem 8.2.6.5. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories. The following conditions are equivalent:
- $(1)$
The coupling $\lambda $ is balanced.
- $(2)$
The coupling $\lambda $ is representable by an equivalence of $\infty $-categories $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$.
- $(3)$
The coupling $\lambda $ is corepresentable by an equivalence of $\infty $-categories $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$.
Proof of Theorem 8.2.6.5.
We will prove the equivalence $(1) \Leftrightarrow (2)$; the equivalence $(1) \Leftrightarrow (3)$ follows by a similar argument. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ is a coupling of $\infty $-categories which is representable by a functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$. Combining Theorem 8.2.5.1 with Proposition 8.2.6.8, we see that $\lambda $ is balanced if and only if the following conditions are satisfied:
The functor $G$ is fully faithful.
The functor $G$ admits a left adjoint $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$.
The functor $F$ is fully faithful.
It follows from Corollary 6.2.2.19 that these conditions are satisfied if and only if $G$ is an equivalence of $\infty $-categories.
$\square$