 Finitary abelian category

A cocomplete abelian category is finitary in the directed-union sense if every cocone under a diagram indexed by a nonempty directed poset, with all its legs monomorphisms, induces a monomorphism from the colimit to the cocone vertex. The category of abelian groups has this property: two colimit representatives mapped to the same vertex become equal at a common upper index because that cocone leg is injective. This use of finitary concerns directed unions, rather than the preservation-of-filtered-colimits meaning used for finitary monads.