Etingof et. al. define abelian categories as additive categories in which for every morphism $\phi : X \to Y$ there exists a sequence $$K \xrightarrow{k} X \xrightarrow{i} I \xrightarrow{j} Y \xrightarrow{c} C$$ with the following properties
- $ji = \phi$,
- $(K,k) = \ker \phi, (C,c) = \operatorname{coker} \phi$,
- $(I,i) = \operatorname{coker} k, (I,j) = \ker c$.
Why is this equivalent to the standard definition of an abelian category: an additive category is abelian if
every map has kernels and cokernels,
every monomorphism is the kernel of its cokernel,
every epimorphism is the cokernel of its kernel.
I assume this is a standard result in some book/paper but I can't find it anywhere.