# Category Theory/Abelian categories

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**Proposition (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 .

**Proof:** 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