Kerodon

Definition 8.2.1.1. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories and let $C$ be an object of $\operatorname{\mathcal{C}}$, having image $\lambda (C) = (C_{-}, C_{+} ) \in \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$. We say that $X$ is universal if it is an initial object of the $\infty $-category $\operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}_{+} } \{ C_{+} \} $, and couniversal if it is an initial object of the $\infty $-category $\{ C_{-} \} \times _{ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{C}}$.