  Note that we have defined \(a\) product for \(X\) and \(Y\) and not the product for \(X\) and \(Y.\) Indeed, products and other objects with a universal property are defined only up to a (unique) isomorphism. Thus in category theory, the nature of the elements constituting a certain construction is irrelevant. What matters is the way an object is related to the other objects of the category, that is, the morphisms going in and the morphisms going out, or, put differently, how certain structures can be mapped into a given object and how a given object can map its structure into other structures of the same kind.