monic epics

Hi, I've been trying to learn some category theory and I came upon the example of a monic, epic in the category of monoids given by the inclusion function of (N,0,+) into (Z,0,+). I know that in monoids every monic arrow is also an injective function but the inclusion function of N into Z provides a counterexample of every epic arrow being a surjective function. I noticed that N is just a "folded" version of Z, where by "folded" I mean take Z and throw away all the inverses of the natural numbers. So does every monic, epic arrow determine such a "folding" or are there monic, epics that can't be characterized in such a way?