question on functors adjoint to their dual

I am interested in the following situation: a contravariant functor adjoint to its own dual, with the unit and counit being the same morphism, but _not_ an iso. The canonical example is the contravariant internal hom on a cartesian (or just symmetric monoidal) closed category, [(_) -> A] for some object A. My question is: is this typical, or are there (interesting) examples of such adjunctions that do not come from exponentials? Thanks, Hayo Thielecke