Remark 8.3.2.8. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a covariant transport representation for $\lambda $. Let $C$ be an object of $\operatorname{\mathcal{C}}$ and set $\lambda (C) = (X,Y)$. Then the isomorphism class of $C$ (as an object of the fiber $\lambda ^{-1} \{ (X,Y ) \} $) can be identified with a connected component $[\eta ]$ of the Kan complex $\mathscr {K}(X,Y )$. Invoking Proposition 5.6.6.21, we deduce the following:
The object $C \in \operatorname{\mathcal{C}}$ is universal (in the sense of Definition 8.2.1.1) if and only if the vertex $\eta \in \mathscr {K}(X,Y)$ is universal (in the sense of Definition 8.3.2.7).
The object $C \in \operatorname{\mathcal{C}}$ is couniversal (in the sense of Definition 8.2.1.1) if and only if the vertex $\eta \in \mathscr {K}(X,Y)$ is couniversal (in the sense of Definition 8.3.2.7).