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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 $\mathcal{C}$. – Rankeya Nov 29 '12 at 2:23
This isn't a subcategory of $C$, it's a subcategory of the arrow category of $C$. – Qiaochu Yuan Nov 29 '12 at 2:54
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. – Zhen Lin Nov 29 '12 at 7:56

The definition is fine in the sense that it gives a well defined mathematical object. Simply put, it is the full subcategory of the category of arrows on $\mathcal {C}$ spanned by the epimorphisms in $\mathcal {C}$. The same will work with any chosen subclass of the arrows on $\mathcal {C}$. Whether or not this is a sensible construction to make depends on the reason why you want to turn the epimorphisms into objects in a category.

Notice that the resulting category is not a subcategory of $\mathcal {C}$.

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