Most general definition of homomorphism and isomorphism What is the most general definition of homomorphism and isomorphism?  It is clear what they mean when there is an algebraic structure to be preserved, but what about when there is no such structure?  Are homomorphism and isomorphism still relevant concepts?
For example, the statement "the real numbers are isomorphic to the equivalence class of Cauchy sequences in the completion of the rational numbers" means what exactly?  Does such a statement presuppose the normal definitions of + and * on the reals and on Cauchy sequences?
In other words, are homomorphism and isomorphism very general concepts that can be used without specifying the operations, or not?

Another candidate for the most general definition is that a homomorphism is just a map between objects of any category, and an isomorphism an invertible map. This is enormously abstract and goes beyond algebra to encompass essentially every type of mathematical object. For that reason, the logic-based answer might be better for your purposes at this point-but in particular for isomorphism the categorical notion is conceptually quite valuable.
