Let $X$ and $T$ be schemes and assume we have two coherent sheaves $\mathcal{F}$ and $\mathcal{G}$ on $X\times T$ which are flat over $T$, that is these are families of sheaves parametrized by $T$.

Assume we have a nontrivial morphism of sheaves $\phi: \mathcal{F}\rightarrow \mathcal{G}$ and that there is some $t_0\in T$ such that $\phi_{t_0}: \mathcal{F}_{t_0}\rightarrow G_{t_0}$ is surjective. Can we find an open subset $U\subseteq T$ containing $t_0$, such that for all $t\in U$ the map $\phi_t$ is surjective?

What can we say in general about the following set: $Z:=\{t\in T | \phi_t: \mathcal{F}_t\rightarrow \mathcal{G}_t \text{is surjective}\}$? Is it always an open subset in $T$? Or do wee need to put some restrictions to $\mathcal{F}$ and $\mathcal{G}$ to be true?

I read in Geometric Invariant Theory and Decorated Principal Bundles by Schmitt: "The condition that a morphism between vector bundles be surjective is an open condition in a suitable parameter space." But nothing more is said about what this "parameter space" should be.