monic epics

[David Karapetyan <dkarapetyan@ucdavis.edu>]

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?