Paul Levy writes:
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?
Yes, and constructing this adjoint equivalence is an incredibly fun exercise in playing around with diagrams for 2-morphisms in your 2-category! The proof must appear in the literature, but I don't know where, and it's really much better to do this sort of thing oneself. Knowing that it's possible should give you the gumption to do it. But if you get stuck, you can find the basic trick in the proof of Prop. 27 in my paper "Higher-dimensional algebra II: 2-Hilbert Spaces", which is available at http://xxx.lanl.gov/abs/q-alg/9609018 Ignore the rather complicated context and just stare at the formulas. Best, jb