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Show if $e : B \rightarrow C$ is the coequalizer of two parallel morphisms $f,g: A \rightarrow B$
and $w: W \rightarrow A$ is epic, then $e$ is the coequalizer of parallel morphisms $fw,gw: W \rightarrow A$

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This is very easy. You should try this yourself. If you have tried, but could not solve it, then you should at least explain where you got stuck. As with just about everything in category theory, you can completely deal with this by drawing a diagram and make a simple argument. – Stefan Hamcke Jun 9 '13 at 14:50
@StefanH. I have never done anything with Category Theory before, honestly I was just stumped and didn't know how to approach the question. I'm sorry if you felt I hadn't made any effort. – sarah jamal Jun 9 '13 at 15:05

Evidently $efw=egw$. Next let $e'fw=e'gw$ for some $e':B\to C'$. Since $w$ is epic it implies $e'f=e'g$. Since $e$ is a coequalizer, there is unique $k:C\to C'$ such that $ke=e'$ as required.

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