= Solution
Let $f:X\to Y$ be a surjective <morphism of schemes> and let $Y'\to Y$ be any <morphism of schemes>. Take any point $y'\in Y'$ 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 $X\times_YY'$ projecting to both $x$ and $y'$. Thus every point of $Y'$ is hit by the projection.
\b[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.
Back to article page
