Example 8.2.1.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ be the twisted arrow coupling of Example 8.2.0.2. For every morphism $f: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$, Corollary 8.1.2.21 asserts that the following conditions are equivalent:
- $(1)$
The morphism $f$ is an isomorphism in $\operatorname{\mathcal{C}}$.
- $(2)$
As an object of $\operatorname{Tw}(\operatorname{\mathcal{C}})$, $f$ is couniversal with respect to the coupling $\lambda $.
- $(3)$
As an object of $\operatorname{Tw}(\operatorname{\mathcal{C}})$, $f$ is universal with respect to the coupling $\lambda $.
In particular, the coupling $\lambda $ is both representable and corepresentable.