TITLE: Epimorphism and Monomorphism = Isomorphism?
QUESTION [30 upvotes]: It seems to be that if a map is both an epimorphism and a monomorphism, it is not necessarily the case that it is an isomorphism.
However, in the category of sets, if a map is both an epimorphism and a monomorphism it is an isomorphism.
Are these accurate statements?

REPLY [13 votes]: Yes, that's correct. For example, every morphism in a poset is a monomorphism and an epimorphism, but only the identity morphisms are isomorphisms. (Mariano's answer describes the smallest poset with a non-identity morphism.) 
However, there are various salvages of this statement available:

A morphism which is both a monomorphism and a split epimorphism (resp. both an epimorphism and a split monomorphism) is an isomorphism. See this blog post. In the category of sets, every monomorphism (edit: with non-empty domain) is split, and the statement that every epimorphism is split is equivalent to the axiom of choice. 
Better: a morphism which is both a monomorphism and a regular epimorphism (resp. both an epimorphism and a regular monomorphism) is an isomorphism. See this blog post. In the category of sets, every epimorphism and monomorphism is regular (with no need to invoke the axiom of choice). 

For more on the subject of morphisms which are both epimorphisms and monomorphisms but which are not isomorphisms see this blog post.