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Let $V,W$ be fixed $k$-vector spaces. Let $C$ be the category whose objects are linear maps $f:V\to W$ and morphisms from $f$ to $g$ are pairs of linear maps $(\alpha,\beta)$ where $\alpha:V\to V,\beta :W\to W$ such that $g \circ\alpha = \beta \circ f$.

What are the epimorphisms and monomorphisms in this category? I tried to prove that if $(\alpha,\beta)$ is an epimorphism, then both $\alpha,\beta$ should be surjective, but failed to prove it.

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This is a strange category (as can be already seen from your other questions about that category). Why are you interested in epis/monos of it? –  Martin Brandenburg Oct 15 '13 at 8:14

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