I am reading the first several pages of GIT by Mumford, and I need some clarification of one requirement in the definition of geometric quotient (c.f. Definition 0.4, GIT):
Suppose a group scheme $G/S$ acts on scheme $X/S$ by $\sigma$, where $G,X$ are schemes over $S$. If a pair $(Y, \phi)$ consisting of a scheme $Y$ over $S$ and an $S-$morphism $\phi: X \to Y$ is a geometric quotient, then one requirement is " the fundamental sheaf $\mathcal{O}_Y$ is the subsheaf of $\phi_*(\mathcal{O}_X)$ consisting of invariant functions" i.e.
If $f \in \Gamma(U,\phi_*(\mathcal{O}_X))= \Gamma(\phi^{-1}(U),\mathcal{O}_X)$, then $f \in \Gamma(U, \mathcal{O}_Y)$ if and only if:
$\begin{matrix} G \times \phi^{-1}(U)&\stackrel{\sigma}{\longrightarrow}&\phi^{-1}(U)\\ \downarrow{p_2}&&\downarrow{F}\\ \phi^{-1}(U)&\stackrel{F}{\longrightarrow}&\mathbb{A}^1 \end{matrix} $ commutes (where $F$ is the morphism defined by $f$, and $\mathbb{A}^1 = \operatorname{Spec}(\mathbb{Z}[x])$)
My questions is how to make sense of this $F$ ?