  Given a map $\phi: X \to Y$ of $\mathbb Q$-factorial Mori dream spaces, one
can ask whether this map is induced by a homogeneous homomorphism $R(Y) \to
R(X)$ of Cox rings. As soon as $Y$ is singular, such a homomorphism needs not
to exist, as pulling back Weil divisors is not well-defined. In this article,
we prove that there is a unique Cox lift $\Phi: \mathcal X \to \mathcal Y$ of
Mori dream stacks coming from a homogeneous homomorphism $R(Y) = R(\mathcal Y)
\to R(\mathcal X)$, where $\mathcal Y$ is a canonical stack to $Y$ and
$\mathcal X$ is obtained from $X$ by root constructions. Moreover, $\phi$ is
induced from $\Phi$ by passing to coarse moduli spaces.
