It looks like it is the uniqueness part I was missing all the time.

I have found the proof in here 3xw.tac.mta.ca/tac/reprints/articles/3/tr3.pdf (3xw = www)

It is lucid, however not direct (not in a way I think about abelian category) so i will present my version of this proof (which is more complicated)

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Step 0

Let us start with , let denote cokernel of f.

We know that f factors through and .

We can do the same with f'. Namely let denote cokernel of f'. Then f' factors through and .

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Step 1

There is a monomorphism

Proof:

Let . Then we can see that there is a map e such that .

We can see that . Hence factorize through m. So we have .

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Step 2

Since is monomorhpism we have .

Next we see that so we can see that . Therefore is an isomorphism.

Finally because every epi is cokernel of its kernel in abelian category.

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My remarks.

I wouldn't be able to do this proof without the book. Their aproach is more lucid, however it is not so beautyful in my opinion.

They argument that there should be not proper subobject in because image can be thought as the smallest subobject that factors f.

It is a good idea to think about image that way. Conversely cokernel can be thought as the biggest quotient object.

Also it is a good idea to think about Ker and Coker as inverse maps on quotient objects and subobjects. And what is important they are order reversing (step one above is the construction of that).