Remark 8.2.1.2 (Uniqueness). Let $\lambda = (\lambda _{+}, \lambda _{-}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories, let $C$ be a universal object of $\operatorname{\mathcal{C}}$, and let $D$ be another object of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
The object $C$ is isomorphic to $D$ (as an object of the $\infty $-category $\operatorname{\mathcal{C}}$).
The object $D$ is universal, and $\lambda _{+}(D)$ is isomorphic to $\lambda _{+}(C)$ (as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{+}$).