Maps of Mori dream spaces

Let ϕ:X→Y be a map of Q-factorial Mori dream spaces. We prove that there is a unique Cox lift Φ:X→Y of Mori dream stacks coming from a homogeneous homomorphism R(Y)=R(Y)→R(X), where Y is a canonical stack of Y and X is obtained from X by root constructions, and ϕ is induced from Φ by passing to coarse moduli spaces. We also apply this techniques to show that a Mori dream quotient stack is obtained by roots from its canonical stack.