Isomorphisms of categories

DeTeXing an exercise I routinely assign, here is an example of an isomorphism of categories that is not `accidental' in Peter Johnstone's sense and is always used in practice as an isomorphism and not merely an equivalence. The fundamental theorem of Galois theory: Let G = Gal(E/F) be the Galois group of a finite Galois extension E/F. Define an isomorphism of categories between the category of intermediate fields F\subset K\subset E and field maps K >--> L that fix F pointwise and the category of orbits G/H and G-maps between them. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]