= Directed colimits created by the abelian-group forgetful functor
{title2=$U:\mathbf{Ab}\to\mathbf{Set}$}
For a diagram of <abelian groups> indexed by a nonempty directed <poset>, take the set colimit as classes $[i,x]$, where $[i,x]=[j,y]$ when their images agree at some common later index. Define $[i,x]+[j,y]=[k,D_{ik}x+D_{jk}y]$ at any common upper bound $k$. Directedness and functoriality prove independence of choices. Negation is $-[i,x]=[i,-x]$ and zero is the class of any zero. This is the unique group structure making the canonical maps homomorphisms, and every compatible group cocone factors by a unique homomorphism. Hence the <forgetful functor> creates these <colimits>, even when transition maps are not injective.
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