Kerodon

Notation 7.6.1.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\{ Y_ i \} _{i \in I}$ be a collection of objects of $\operatorname{\mathcal{C}}$. We will say that an object $Y \in \operatorname{\mathcal{C}}$ is a product of the collection $\{ Y_ i \} _{i \in I}$ if there exists a collection of morphisms $\{ q_ i: Y \rightarrow Y_ i \} $ which exhibits $Y$ as a product of $\{ Y_ i \} _{i \in I}$. If this condition is satisfied, then the object $Y$ is uniquely determined up to isomorphism (see Proposition 7.1.1.12). To emphasize this uniqueness, we will sometimes denote the object $Y$ by ${\prod }_{i \in I} Y_ i$, and refer to it as the product of the collection $\{ Y_ i \} _{i \in I}$. Similarly, we say that $Y$ is a coproduct of the collection $\{ Y_ i \} _{i \in I}$ if there exists a collection of morphisms $\{ e_ i: Y_ i \rightarrow Y \} _{\in I}$ which exhibits $Y$ as a coproduct of $\{ Y_ i \} _{i \in I}$. In this case, we sometimes denote the object $Y$ by ${\coprod }_{i \in I} Y_ i$ and refer to it as the coproduct of the collection $\{ Y_ i \} _{i \in I}$.