Saturated base change of toroidal morphisms

We show that saturated base change of a dominant toroidal morphism is also toroidal. For completeness, we give full details on equivalence between definitions regarding toroidal embeddings and toroidal morphisms in literature. Moreover, we show in detail the equivalence between toroidal embeddings and logarithmically regular log-varieties, respectively, toroidal morphisms and logarithmically smooth morphisms.