Let $S$ be a scheme.

**Definition.** Let $X$ be an $S$-scheme and $G$ a smooth affine group $S$-scheme acting on $X.$ An $S$-scheme $Y$ is a *geometric quotient* of $X$ by $G$ if there exists a morphism $\pi_X\colon X\rightarrow Y$ such that

$\pi_X$ is $G$-invariant,

the geometric fibers of $\pi_X$ are orbits,

$\pi_X$ is universally submersive, i.e., $U\subset Y$ is open iff $\pi_X^{-1}(U)\subset X$ is open, and this property is preserved by base change,

$(\pi_X)_*(\mathcal{O}_X)^G=\mathcal{O}_Y.$

Let $X$ and $X'$ be $S$-schemes with a $G$-action, where $G$ is the same introduced before. Assume that there exist geometric quotients $\pi_X\colon X\rightarrow Y$ and $\pi_{X'}\colon X'\rightarrow Y'.$

**Question.** Let $g\colon Y\rightarrow Y'$ be a morphism. Is there a $G$-equivariant morphism $f\colon X\rightarrow X'$ such that $\pi_{X'}\circ f = g\circ \pi_{X}$?