If A' is isomorphic to A'' and B' is isomorphic to B'' then A'UB' is isomorphic to A''UB''. (their intersections are empty).
Moreover conclude that the disjoint union is well-defined up to isomorphism.
Can you use results about equivalence classes and partitions for this result? (i.e. show that disjoint equivalence classes have a natural extension to isomorphism results)