  Category theory unifies mathematical structures in two different ways. First, as we have seen, almost every set theoretically defined mathematical structure with the appropriate notion of homomorphism yields a category. This is a unification provided within a set theoretical environment. Second, and perhaps even more important, once a type of structure has been defined, it is imperative to determine how new structures can be constructed out of the given one. For instance, given two sets \(A\) and \(B\), set theory allows us to construct their Cartesian product \(A \times B.\) It is also imperative to determine how given structures can be decomposed into more elementary substructures. For example, given a finite Abelian group, how can it be decomposed into a product of certain of its subgroups? In both cases, it is necessary to know how structures of a certain kind may combine. The nature of these combinations might appear to be considerably different when looked at from a purely set theoretical perspective. 