# Is the requirement that every morphism factors as an epi composed with a mono part of the definition of an abelian category?

Hilton and Stammbach require an abelian category to be an additive category in which

1) all kernels and cokernels exist

2) all monos are the kernel of their cokernel, all epis are the cokernel of their kernel

3) every morphism factors as an epi composed with a mono

But Wikepedia's definition only requires it to be an additive category in which

1) all kernels and cokernels exist

2') all (co)kernels are (co)normal.

I managed to get 2') from 2), so they are equivalent. Now I'm wondering how to get a 3) from 1) and 2) but I'm stuck. Even Hilton and Stammbach made explicit use of 3) when they were proving their Proposition 9.6:

In an abelian category any morphism $\phi : A\rightarrow B$ factors as $K \hookrightarrow A \twoheadrightarrow I \hookrightarrow B \twoheadrightarrow C$.

and I can't see a way to get around it and prove 3) from the other two.

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In 2') you mean that monos/epis are normal/conormal. – Martin Brandenburg Oct 27 '13 at 20:04
The proof of 3) (which follows from 1,2) can be found in the book CWM by Mac Lane. – Martin Brandenburg Oct 27 '13 at 20:05
possible duplicate of Some questions on abelian category – Zhen Lin Nov 2 '13 at 17:05