 Solution

Let $f:X\to Y$ be a surjective morphism of schemes and let $Y^{\prime }\to Y$ be any morphism of schemes. Take any point $y^{\prime }\in Y^{\prime }$ and let $y$ be its image in $Y$. Surjectivity supplies $x\in X$ with $f(x)=y$. By the point-lifting property of a scheme fibre product proved in part (c), there is a point of $X{\times }_Y{Y}^{\prime }$ projecting to both $x$ and $y^{\prime }$. Thus every point of $Y^{\prime }$ is hit by the projection.

This proves that surjectivity is preserved by base change. No restriction is imposed on the base change of a morphism of schemes. No flatness or finite-type assumption is required.