Remark 8.2.1.4 (Symmetry). Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories, and let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\operatorname{\mathcal{C}}_{+}$ with $\operatorname{\mathcal{C}}_{-}$. Then the transposition $\lambda ' = ( \lambda _{+}, \lambda _{-} )$ can be regarded as a coupling of $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}}$ with $\operatorname{\mathcal{C}}_{+}^{\operatorname{op}}$. In this situation:
An object $C \in \operatorname{\mathcal{C}}$ is universal for the coupling $\lambda $ if and only if it is couniversal for the coupling $\lambda '$.
The coupling $\lambda $ is representable if and only if the coupling $\lambda '$ is corepresentable.