# Exercise in an abelian category

Supose we have an exact sequence $$A\overset{f}\longrightarrow B\overset{g}\rightarrow C\overset{h}\rightarrow D$$ in an abelian category $\mathcal{A}$. Is it true that $f$ is an epimorphism if and only if $h$ is a monomorphism?

It is clear that for any category of modules this is true.

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$f$ epi $\Leftrightarrow$ $\mathrm{im}(f)=B$ $\Leftrightarrow$ $\ker(g)=B$ $\Leftrightarrow$ $g=0$ $\Leftrightarrow$ $\mathrm{im}(g)=0$ $\Leftrightarrow$ $\ker(h)=0$ $\Leftrightarrow$ $h$ mono.
Let $f$ be an epimorphism. If the image of $f$ is the kernel of $g$ then $g$ is constant and then the image of $g$ being the kernel of $h$ is trivial which is enough to have $h$ as an monomorphism.
Let $h$ be a monomorphism, then the image of $g$ is constant, hence $f$ is surjective.
"the image of $g$ is constant" and "$f$ is surjective" are not precise. – Martin Brandenburg Feb 21 '14 at 8:53