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Here is how I thought about it:

Suppose $\mathcal{C}$ is a category and $\mathcal{E}$ is the subclass of all epimorphisms of $\mathcal{C}$. I am thinking to a subcategory of $\mathcal{C}$, which has all of epimorphisms as its objects and all commuting squares as its morphisms (just like the construction of arrow category).

Is everything fine with such a definition?

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    In this case I would not call $\mathcal{E}$ a subcategory of $\mathcal{C}$.2012-11-29
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    This isn't a subcategory of $C$, it's a subcategory of the arrow category of $C$.2012-11-29
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    Of course, you could also form the non-full subcategory of $\mathcal{C}$ whose morphisms are the epimorphisms. This makes sense because the composite of two epimorphisms is another epimorphism.2012-11-29

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