Abelian Closures
January-14-11
8:42 AM
To be more explicit, the following universal property can be used. A kernel of f is any morphism k : K → X such that:
f o k is the zero morphism from K to Y;
Given any morphism k′ : K′ → X such that f o k′ is the zero morphism, there is a unique morphism u : K′ → K such that k o u = k'.
Pasted from <[link])> 
Explicitly, this means the following. The cokernel of f : X → Y is an object Q together with a morphism q : Y → Q such that the diagram
commutes. Moreover the morphism q must be universal for this diagram, i.e. any other such q′: Y → Q′ can be obtained by composing q with a unique morphism u : Q → Q′:
Pasted from <[link]> 