Kerodon

Remark 7.7.4.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\{ X_ i \} _{i \in I}$ be a collection of objects of $\operatorname{\mathcal{C}}$. If there exists a collection of morphisms $\{ f_ i: X_ i \rightarrow X \} _{i \in I}$ which exhibits $X$ as a coproduct of $\{ X_ i \} _{i \in I}$, then the collection $\{ f_ i \} _{i \in I}$ is uniquely determined up to isomorphism (as an object of the $\infty $-category $\operatorname{\mathcal{C}}\times _{ \prod _{i \in I } \operatorname{\mathcal{C}}} \prod _{i \in I} \operatorname{\mathcal{C}}_{/X_ i}$). Consequently, conditions conditions $(1)$ and $(2)$ of Definition 7.7.4.1 depend only on the collection $\{ X_ i \} _{i \in I}$, and not on the choice of the morphisms $f_ i$.