Dear Jonathan, I think you misread Mac Lane. If the 2-category is Cat , suppose functors f: A --> B and u: B --> A are such that fu and uf are isomorphic to id(A) and id(B) by isomorphisms eta: id(A) -->uf and epsilon: fu --> id(B). He does not say that one can modify eta and epsilon can be to satisfy the the triangle identities. What he says is: You can keep f and modify u, eta and epsilon to get a functor u': B ---> A and isomorphisms eta': id(A) ---> u'f and epsilon': fu' ---> id(B) satisfying the triangle identities. Under this form the result is true for any 2- Category C. Sketch of proof: Suppose f: A --> B and u: B ---> A are 1-cells and eta and epsilon are isomorphism 2-cells. Then for each object X of C, f,u,eta and epsilon define an equivalence between the categories C(X,A) and C(X,B). Take X=B, then f defines an equivalence C(B,f): C(B,A) --> C(B,B). By Mac Lane's result we can find an adjoint to C(B,f), which is en equivalence, U: C(B,B) ---> C(B,A). Take u' = U(id(B)): B --->A . By a tedious but straightforward computation one verifies that the pair (f, u') is an adjoint equivalence. Bien amicalement, Jean Le 6 avr. 2013 à 22:22, Jonathan CHICHE 齊正航 a écrit :
Dear all,
This is a standard fact, proved for instance in details in "Categories for the Working Mathematician" (Chapter 4, Section 4, Theorem 1 in the second edition), that a functor is an equivalence of categories if and only if it is part of an adjoint equivalence. I would like to use the fact that this is true in an arbitrary 2-category, i.e. that given an equivalence in a 2-category the invertible 2-cells can be required to satisfy the triangle identities. Is there a standard reference for this fact? Well-known books such as classical introductions to category theory are particularly welcome.
Thanks,
Jonathan
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