TITLE: Is category equivalence unique up to isomorphism?
QUESTION [7 upvotes]: Let  $\mathbf C$, $\mathbf D$ be categories and $F, F':\mathbf C \to \mathbf D$ and $G, G':\mathbf D \to \mathbf C$ be functors of the shown direction.  Is it the case that, if each of $(F,G)$ and $(F',G')$ is an equivalence, then $F\cong F'$ (isomorphic in $\mathbf D^\mathbf C$)?
Initially I thought (without thinking) that there couldn't be more than "one" equivalence (up to isomorphism) between two categories.  But, once I tried to prove this, I couldn't find any  reason whay it should be so.  Was I wrong?  Any help will be appreciated.

REPLY [2 votes]: Peter Freyd's book "Abelian Categories" contains a discussion of this topic in Exercise B at the end of Chapter 1. He defines the automorphism class group of a category $C$ to be the group of equivalences $C\to C$ modulo those naturally isomorphic to the identity.  As an example where this is non-trivial, he gives the category of ordered sets and order-preserving functions.  The functor that reverses the ordering is an equivalence but is not isomorphic to the identity.