Definition 8.2.6.1. Let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories. We say that $\lambda $ is balanced if it satisfies the following conditions:
- $(1)$
The coupling $\lambda $ is representable. That is, for each object $C_{+} \in \operatorname{\mathcal{C}}_{+}$, there exists a universal object $C \in \operatorname{\mathcal{C}}$ satisfying $\lambda _{+}(C) = C_{+}$.
- $(2)$
The coupling $\lambda $ is corepresentable. That is, for each object $C_{-} \in \operatorname{\mathcal{C}}_{-}$, there exists a couniversal object $C \in \operatorname{\mathcal{C}}$ satisfying $\lambda _{-}(C) = C_{-}$.
- $(3)$
An object $C \in \operatorname{\mathcal{C}}$ is universal if and only if it is couniversal.