I've always been surprised by the fact that $(Epi, Mono)$ is, when possible, a strong factorization system (unique solution to lifting problems, existence of unique factorization), whereas $(Mono, Epi)$ is only a weak factorization system. How can one tell when something similar happen in general? Is there any other known example of such a situation?
PS: I know that orthogonality (which is the thing I'm focused on in my question) is not enough to define a factorization system, weak or not. If you think the question is more amenable replacing factorizations with prefactorizations, feel free to consider the problem of determining when $\mathcal E\perp \mathcal M$ (strong) and $\mathcal M\pitchfork\mathcal E $ (weak).
