Definition 7.7.4.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that a collection of objects $\{ X_ i \} _{i \in I}$ have a disjoint coproduct if the following conditions are satisfied:
- $(0)$
There exists an object $X \in \operatorname{\mathcal{C}}$ and a collection of morphisms $\{ f_ i: X_ i \rightarrow X \} $ which exhibit $X$ as a coproduct of the collection $\{ X_ i \} _{i \in I}$ (Definition 7.6.1.3).
- $(1)$
For each $i \in I$, the morphism $f_ i: X_ i \rightarrow X$ is a monomorphism (Definition 9.3.4.1).
- $(2)$
For every pair of distinct elements $i,j \in I$, there exists a pullback $X_ i \times _{X} X_ j$, which is an initial object of $\operatorname{\mathcal{C}}$.