While making a web search I came across an old message by Paul Levy. He had originally asked (in message 200111201650.fAKGol115859@foobar.pps.jussieu.fr <http://north.ecc.edu/alsani/ct01%289-12%29/msg00071.html>):
If P and Q are objects in a 2-category C, and there is an equivalence between them, must there be an adjoint equivalence (an adjunction whose unit and counit are both isomorphisms) between them?"
After the answer was given he revealed the motivation for his question:
I'm trying to make an argument that the natural 2-categorical analogue of isomorphism is adjoint equivalence rather than equivalence, but your result suggests that it doesn 't matter.
I am currently wondering about a closely related observation. While playing around with the notion of 2-transport, I noticed that, contrary to my original assumption, in order for a certain 2-functor to be expressible in terms of another 2-functor (to be "locally trivializable" in my application) it suffices for both 2-functors to be related by a "special ambidextrous adjunction". By a special ambidextrous adjunction I mean an ambidextrous adjunction A --L--> B -- R --> A such that 1 ==> LR ==> 1 and 1 ==> RL ==> 1 are identity 2-morphisms. This is strictly weaker than an adjoint equivalence. (I have chosen the adjective "special" in order to allude to the fact that the Frobenius algebras obtained from these adjunctions are called "special Frobenius algebras".) I would like to know if there are any well known insights concerning such "special ambidextrous adjunctions". P.S. There are notes with more details and examples here: http://www.math.uni-hamburg.de/home/schreiber/FRSfrom2Transport.pdf