Hi, I have a question. 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?
The answer is yes. Let f : Q --> P and u : P --> Q be arrows in a 2-category K with an invertible 2-cell e : f u --> 1 and some invertible 2-cell q : 1 --> u f (some say f is quasi-inverse to u). The existence of q implies that the functor K(X,f) is fully faithful for all objects X of K. We define the unit n : 1 --> u f by the condition that f n should be the inverse of e f (using fullness of K(Q,f)). So one adjunction triangle is satisfied. The other follows by the faithfulness of K(P,f). An application of this is that a pseudonatural transformation (between pseudofunctors) which is a pointwise equivalence has a quasi-inverse pseudonatural transformation. --Ross