$X$: projective scheme over a scheme $S$.

$E, F$: $\mathscr{O}_X$-modules, flat/$S$

$\phi$: $E \rightarrow F$ : morphism s.t. $\phi_t$: $E_t \rightarrow F_t$ is zero morphism for all $t \in S$

Then, is $\phi$ zero morphism ?

I'd be glad if you could tell me something! (Please give me some comments about the comment below!)

Edit: especially I am interested in the case $X = Y \times S$ ,where $Y$: projective surface / $\mathbb{C}$, $S: \mathbb{C}$-scheme