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This is a homework question. I have already searched the web, but the proofs I have found are very generalized using category theory. Problem is the following:

Suppose you have homomorphism $f: M \to N$ of $R$-modules und $\pi$ is the canonical projection of $N$ on the cokernel of $f$.
Now one has to show the universal property of the cokernel, that is $\forall$ $R$-homomorphisms $g : N \to P$ of modules with $g \circ f = 0$ exists a unique homomorphism $\tilde g: \mathrm{coker}(f) \to P$ with $\tilde g \circ \pi = g$ and show that the pair $(\mathrm{coker}(f), \pi)$ is unique up to unique isomorphism.


Showing the existence of the universal property is not hard. I want to show the pair is unique up to unique isomorphism.
My approach: Let $(D, \delta)$ be another pair satisfying the universal property. As $\pi \circ f = 0$, there exists unique morphism $\hat g : D \to \text{coker}(f)$ with $\hat g \circ \delta = \pi$.
Now if i could show that $\delta \circ f = 0$, I would get another morphism from the cokernel to $D$, and I can show that $\tilde g$ is the desired unique isomorphism.
But is this the right approach? Are there any other maps I should take a look at and use the universal property on?

I would prefer hints over full solutions, but both is fine of course.
Thanks!

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    $\begingroup$ $\delta\circ f=0$ is part of the characterisation of the universal property in this case. $\endgroup$
    – Bernard
    Apr 25, 2016 at 18:16
  • $\begingroup$ Okay, thanks, this was not clear to me! $\endgroup$
    – johnnycrab
    Apr 25, 2016 at 18:55

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