In this talk we will discuss a paper by Florian Berchtold which determines when a morphism of toric varieties lifts to a quotient presentation; the Cox construction furnishes one (special!) quotient presentation, but there are other possibilities. It turns out that this question is intimately related to pulling back Weil divisors along the morphism being studied. We will discuss this relationship and then discuss Berchtold's proof that two toric varieties which are isomorphic as abstract varieties are isomorphic as toric varieties.