Surjectivity is preserved by base change
= Surjectivity is preserved by base change
If $X\to Y$ is a surjective <morphism of schemes>, the projection $X\times_YY^{\prime}\to Y^{\prime}$ is surjective for every $Y^{\prime}\to Y$. For a point $y\prime$, choose a point $x$ over its base image and apply the <point-lifting property of a scheme fibre product>. No flatness assumption is necessary.