Let $\mathcal A$ be a category and $\bar{\mathcal{E}}$ be the largest subset of all Epis, stable under composition, such that every mono in $\bar{\mathcal{E}}$ is iso.
What is known about $\bar{\mathcal{E}}$?
Obviously every strong epi is in $\bar{\mathcal{E}}$ (assuming these are actually epi). What can be said about the converse, is there a counterexample?
If so, do the elements of $\bar{\mathcal{E}}$ have a name? Are there any other known properties or characterizations in certain kinds of categories?