It’s a curious (though well known) phenomenon that an equivalence in a bicategory can always be converted into an adjoint equivalence by tweaking one of the 2-cells. There are two ways to prove it, that I know of. The elementary way is just to write down a (rather complicated) definition and laboriously prove that it has the desired properties. I always found this rather baffling, so I preferred the more abstract approach: in essence, it’s easy to show that equivalences in Cat have this property by using the Yoneda lemma, then one can use the bicategorical Yoneda lemma to transfer the result to an arbitrary bicategory. But recently I learnt something that makes the direct approach a lot easier for me to understand.
At the CMS meeting in Calgary I gave a talk about monoidal categories in which I described how, under certain circumstances, the coherence conditions for a monoidal category are redundant. What I meant is that if you give me any natural isomorphisms
αA,B,C: A⊗(B⊗C) → (A⊗B)⊗C
λA: I⊗A → A
ρA: A⊗I → A
then I can use them to construct new ones that satisfy the coherence conditions. This doesn’t always work – my argument assumes a technical condition called tensor generation – but a key part of the construction is perfectly general. Here it is: suppose we have a category C equipped with a functor ⊗: C×C → C and a specified object I. If you give me natural isomorphisms
λA: I⊗A → A
ρA: A⊗I → A
then I can construct a new natural isomorphism λ’ with the property that λ’I = ρI. I simply define λ’A to be the composite
(The only trick needed to see that λ’I = ρI is the old observation of Kelly that, since ρ is natural, the square
commutes, and since it’s invertible that implies ρI⊗I = ρI⊗I. Just use that and the naturality of λ.)
After the talk, Robert Paré asked me whether turning an equivalence into an adjoint equivalence was an instance of this construction. Not as far as I know, I said. But his hunch was right, as he showed me the next morning after he’d figured it out. Given an equivalence
in some 2-category B, consider the hom category B(X, Y), define a tensor on it by a⊗b := bga, and take the ‘unit’ object to be f. The λ and ρ are given by the 2-cells fg ⇒ 1 and gf ⇒ 1 from the equivalence. Then you construct λ’, and lo and behold! the equation λ’I = ρI corresponds exactly to one of the triangle equations for an adjunction. (The other can then be derived using the fact that the 2-cells are invertible.) It also works when B is a bicategory: the only real difference is you have to choose one of the bracketings for bga, say b(ga), and put in a couple of associators.