= 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>.
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