I am looking for a counter-example of two functors F : C -> D and G : D->C such that


1) F is left adjoint to G


2) F is right adjoint to G


3) F is not an equivalence (ie F is not a quasi-inverse of G)



Is a functor which has a left adjoint which is also its right adjoint an equivalence ?
The answer of Ben Webster, can be made easier. Consider the functor F : (A-mod) -> (A-mod) which maps any A-module on (0). Then, F is a left adjoint to F ; and so, is a also a right adjoint to F. This is clear because for all A-modules N, M, one has Hom\_A(0,N)=Hom\_A(M,0). But, F is not an equivalence.


