Maps of Mori dream spaces

Let phi: X -> Y be a map of Q-factorial Mori dream spaces. We prove that there is a unique Cox lift Phi: 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 phi is induced from 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.