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Let $$\matrix{ A& \mathop{\longrightarrow}\limits^f &B\\ \Big\downarrow & & \Big\downarrow\\ C&\mathop{\longrightarrow}\limits_g &D }$$ Be a pushout diagram in a category $\mathcal C$. If $f$ is monic, is $g$ also monic?

I have known that this holds in an abelian category. Is it true for a general category? If so, how to prove it?

If if fails, could anyone give me a counterexample? And what conditions should we impose on the category $\mathcal C$ to ensure that this is true?

Thanks!

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    Monics that *are* preserved under all pushouts are called (in French) *monomorphismes universels* (the dual concept, *épimorphismes universels*, is introduced in SGA 4 (I.10.3) along with an extensive list of other properties epimorphisms may or may not have).2012-06-06
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    @t.b.: 1+. And there is a (rather sparse) nlab article on that concept: http://ncatlab.org/nlab/show/universal+epimorphism2012-06-06
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    @Martin: Thanks! I wasn't sure whether that name caught on or not and last time I checked that nlab page didn't exist... Obviously some people still use it :) (+1 for your nice answer, by the way!)2012-06-06

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