Let $f:X \to Y$ be a flat, projective morphism between projective varieties. Let $F, G$ be coherent sheaves on $X$, flat over $Y$. Let $\phi_1, \phi_2$ be two morphisms from $F$ to $G$ such that:

1) the image of $\phi_1$ and $\phi_2$ are flat over $Y$,

2) for every $y \in Y$, $\mathrm{Im}(\phi_1)|_{X_y} \subset G|_{X_y}$ and $\mathrm{Im}(\phi_2)|_{X_y} \subset G|_{X_y}$, where $X_y:=f^{-1}(y)$.

Is it then true that the morphism $\phi_1+\phi_2:F \to G$ has the same properties i.e.,

1) the image of $\phi_1+\phi_2$ is flat over $Y$ and

2) for every $y \in Y$, $\mathrm{Im}(\phi_1+\phi_2)|_{X_y} \subset G|_{X_y}$?