TITLE: Monic and epic implies isomorphism in an abelian category?
QUESTION [8 upvotes]: Is it true that monic and epic implies isomorphism in an abelian category?

REPLY [12 votes]: The answer is yes, as a consequence of two facts. First, in any category (abelian or not), if the equalizer of a pair of maps is epic, then it is an isomorphism. (Proof: By definition of epic, the two maps that it equalizes must be equal, and then an isomorphism clearly serves as the equalizer.) Second, in an abelian category, every monomorphism is the kernel of some map (this is axiom A3 in the definition of abelian category, on page 35 of Freyd's book "Abelian Categories), which means it's the equalizer of that map and a zero map.