"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?" Thanks Richard, Ross and John for your affirmative answer. I would have preferred a counterexample, but that's life. 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. What about automorphism groups? Say we have an object P in a 2-category C. We can either consider the category (strict monoidal with inverses) of auto - adjoint equivalences on P and transformations between them, or we can consider the category (strict monoidal with inverses) of auto - equivalences on P and transformations between them. Must these be equivalent? Thanks Paul -- Paul Blain Levy Universite Paris 7 http://www.pps.jussieu.fr/~levy