Category Theory/Abelian categories

object in abelian category is decomposed into sum by subobject Let be an abelian category, and let be an object. Let be a subobject, and let be the corresponding quotient object. Moreover, denote . Then there exists a unique isomorphism such that and . is a biproduct. First we apply the universal property of a product in order to obtain a morphism such that and . Then we apply the universal property of a coproduct in order to obtain a morphism such that and . Moreover, we get a morphism from the projection to , and a morphism from the inclusion of . The latter morphism is the kernel of , and the cokernel of that kernel is