Implication Details

Assumptions: biproducts, filtered colimits, filtered-colimit-stable monomorphisms, products

Conclusions: CIP

Reason: Let (Xi)i∈I(X_i)_{i \in I} be a family of objects. For every finite subset E⊆IE \subseteq I the canonical morphism ∐i∈EXi=∏i∈EXi→∏i∈IXi\coprod_{i \in E} X_i = \prod_{i \in E} X_i \to \prod_{i \in I} X_i is a (split) monomorphism. Hence, their colimit is also a monomorphism, which is the canonical morphism ∐i∈IXi→∏i∈IXi\coprod_{i \in I} X_i \to \prod_{i \in I} X_i.