Let $G$ be an algebraic group acting on a scheme X, and $\phi:X\rightarrow Y$ be a morphism of schemes. We define the subsheaf of invariants $\mathcal I_\phi$ of $\phi_*\mathcal O_X$ by the fact that for any open subset $U$ of $Y$, a section $f\in\Gamma(U,\phi_*\mathcal O_X)=\Gamma(\phi^{-1}(U),\mathcal O_X)$ belongs to $\Gamma(U,\mathcal I_\phi)$ if and only if the following diagram commutes, where $\sigma$ stands for the action, $p_2$ for the projection on the second factor, and $F$ for the morphism induced by $f$.
I read, in Mumford's GIT book, that if $\mathcal O_Y$ is the subscheaf of invariants of $\phi_*\mathcal O_X$, then $\phi$ is dominating. Why?