Summary: In this article we investigate algebraic morphisms between toric varieties. Given presentations of toric varieties as quotients we are interested in the question when a morphism admits a lifting to these quotient presentations. We show that this can be completely answered in terms of invariant divisors. As an application we prove that two toric varieties, which are isomorphic as abstract algebraic varieties, are even isomorphic as toric varieties. This generalizes a well-known result of A. S. Demushkin [Mosc. Univ. Math. Bull. 37, 104-111 (1982; Zbl 0507.14033)] on affine toric varieties.