Category Theory/Abelian categories

Proposition (object in abelian category is decomposed into sum by subobject):

Let ${\mathcal {C}}$ be an abelian category, and let $X\in {\mathcal {C}}$ be an object. Let $f:Y\to X$ be a subobject, and let $Z=\operatorname {Coker} f$ be the corresponding quotient object. Moreover, denote $g=\operatorname {coker} f$. Then there exists a unique isomorphism $\theta :X\to Y\oplus Z$ such that

$\pi _{Z}\circ \theta =g$ and $\theta ^{-1}\circ \iota _{Y}=f$.

Proof: $Y\oplus Z$ is a biproduct. First we apply the universal property of a product in order to obtain a morphism

$\phi :X\to Y\oplus Z$ such that $\pi _{Z}\circ \phi =g$ and $\pi _{Y}\circ \phi =0$.

Then we apply the universal property of a coproduct in order to obtain a morphism

$\chi :Y\oplus Z\to X$ such that $\chi \circ \iota _{Y}=f$ and $\chi \circ \iota _{Y}=0$.

Moreover, we get a morphism $Y\otimes X\to Y\otimes 0$ from the projection to $Y$, and a morphism $0\otimes Z\to Y\oplus Z$ from the inclusion of $Z$. The latter morphism is the kernel of ${\tilde {f}}:Y\oplus Z\to X$, and the cokernel of that kernel is $\Box$