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In this question the poster says that, if one of the two maps with the same domain is monic, then the corresponding induced map in the pushout diagram is also monic, in an abelian category.

Or, dually, if in the following pull-back diagram
$\matrix{A& \longrightarrow &B\\ f\Big\downarrow & & \Big\downarrow g\\ C&\longrightarrow &D }$
$g$ is epic, then so is $f$.

I tried to employ of the universal properties, which sppear to be the only tools avaliable, but with no luck: more precisely, I could show this if I can show that this pullback is also a pushout, then I can show that $f$ is an epic as well. And this is where I am stuck at.
So any help or reference is welcomed. Thanks in advance.
P.S. Sorry for asking the dual statement of the title. It suits well to prove any of the two dual statements.

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I gave a hint here math.stackexchange.com/questions/637350/… –  Berci Feb 4 at 0:27
    
@Berci Thanks for that. I already found out a proof, inspired by the book by Freyd and the Stack project. Amusingly, my initial goal was to show the first isomorphism theorem in an abelian category. That was why I ought to be careful about the notion of exact sequences. In any case, thanks very much. –  awllower Feb 4 at 14:56
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1 Answer

up vote 3 down vote accepted

The universal properties are not the only tools available. In fact, this stability of epimorphisms under pullbacks does not hold in arbitrary categories (see the linked question). We really have to use the fact that we are in an abelian category. You can find the proof in the standard sources, for example:

S. Mac Lane, Categories for the working mathematician, Prop. 2 in section VIII.4.

P. Freyd, Abelian categories, Prop. 2.54

Stacks project, Tag 08N4

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Thanks for your inspiring references. –  awllower Feb 11 at 14:05
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