The dual notion is an extremal monomorphism: a morphism such that if where is an epimorphism, then is an isomorphism.
Despite the terminology, it is not necessarily true that an extremal epimorphism is in fact an epimorphism, so sometimes this is required as well (and dually for extremal monomorphisms). On the other, this does often follow, as in the first remark below.
Of course, the dual properties are all true of extremal monomorphisms. (See coequalizer, monomorphism, strong monomorphism, pushout, regular monomorphism, coregular category?, coimage factorization?, epimorphism.)