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?