First of all, I know the concepts of isomorphism and equivalence between categories, and that the latter one is the more interesting one, whereas the first is rather rare and uninteresting. 


Are there isomorphisms of categories, which are not trivial and not pathological? I regard the examples on [wikipedia](http://en.wikipedia.org/wiki/Isomorphism_of_categories) as trivial, because these are only reformulations of the definitions of the objects in consideration. Thus perhaps the question is: Are there nontrivial reformulations?


There are lots of nontrivial equivalences of categories (affine schemes <-> rings (dual), compact hausdorff spaces <-> unital commutative C\*-algebras (dual), finite abelian groups <-> finite abelian groups (dual), skeletons such as the algebraic extensions of function fields over fixed prime fields in the category of fields), but I wonder if these categories are actually isomorphic. Of course, in the examples of interest, you can't take the known equivalence as an isomorphism, but perhaps there is another one?



nontrivial isomorphisms of categories
Whether this counts as trivial is a subjective matter, but here goes.


Any adjunction 
$$
F: C \to D,\ \ \ G: D \to C
$$ 
(with $F$ left adjoint to $G$) gives rise canonically to a monad $T = GF$ on $C$ and a "comparison" functor $K: D \to C^T$. Here $C^T$ is the category of algebras for the monad $T$. The adjunction is said to be **monadic** if $K$ is an equivalence of categories.


Now in fact, for most of the obvious examples of monadic adjunctions, the comparison is actually an *isomorphism*. For example, if $G$ is the forgetful functor from groups to sets then it's an isomorphism. The same is true if you replace groups by any other algebraic theory (rings, Lie algebras, etc). 


Indeed, if you look in *Categories for the Working Mathematician*, you'll see that Mac Lane calls a functor monadic if $K$ is an *isomorphism*. He does the whole basic theory of monads with this definition. I suspect this is because $K$ really is an isomorphism in the standard examples. *CWM* was published in 1971, and since then it's become clear that Mac Lane's definition was too narrow. Whether the pioneers of monad theory (such as Beck) also used this narrow definition, I don't know.


