I should have mentioned another quite elementary example, that is perhaps more intriguing. Let us write Top for (topological spaces defined by open sets) and Top' for the isomorphic category (topological spaces defined by closed sets). Let (X, L) be a set X equipped with a complete sublattice L of its lattice of parts. Viewing it as on object of Top or Top' will interchange an Alexandrov topology for X with the opposite one, generally different. This says that - formally - we cannot think of these two isomorphic categories as being the same thing. Even if, of course, we do think that way, informally and in practice. I am not entirely convinced by a comment of Peter: "in practice (so far as I know) one never makes use of the fact that they are isomorphisms rather than mere equivalences". I am happy with the fact that, going from Top to Top' and back, we get the same space on the nose; this spares a lot of complications. Best regards Marco Grandis [For admin and other information see: http://www.mta.ca/~cat-dist/ ]