categories: Re: adjoint equivalence

[Ross Street <street@ics.mq.edu.au>]

>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