Kerodon

Variant 8.2.1.6. Let $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ be a functor of $\infty $-categories, set $\operatorname{\mathcal{C}}= \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}) \times _{ \operatorname{\mathcal{C}}_{-} } \operatorname{\mathcal{C}}_{+}$, and let $\lambda _{G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ denote the coupling of Construction 8.2.0.3. Unwinding the definitions, we see that objects of $\operatorname{\mathcal{C}}$ can be identified with pairs $(e, C_{+} )$, where $C_{+}$ is an object of the $\infty $-category $\operatorname{\mathcal{C}}_{+}$ and $e: C_{-} \rightarrow G(C_{+})$ is a morphism in the $\infty $-category $\operatorname{\mathcal{C}}_{-}$. It follows from Example 8.2.1.5 that an object $(e,C_{+}) \in \operatorname{\mathcal{C}}$ is universal if and only if $e$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}_{-}$. Note that every object $C_{+} \in \operatorname{\mathcal{C}}_{+}$ can be lifted to a universal object of $\operatorname{\mathcal{C}}$ (for example, we can choose $e$ to be the identity morphism from $G(C_{+})$ to itself), so that the coupling $\lambda _{G}$ is representable. In §8.2.3, we will prove the converse: every representable coupling of $\infty $-categories can be obtained (up to equivalence) from Construction 8.2.0.3(Theorem 8.2.0.4).