Is every morphism of $K$-schemes a $K$-morphism?

Question

Let $S$ be a scheme, let $X$ and $Y$ be $S$-schemes, and let $f:X \to Y$ be a morphism of schemes.

I was wondering whether it is safe to assume that $f$ is an $S$-morphism, i.e. whether $f$ fits into a commutative triangle with the structural morphisms $X \to S$ and $Y \to S$.

Is the case $S= \operatorname{Spec}K$, with $K$ a field, special in any way?

Answer 1

Complex conjugation on the coefficients defines a ring automorphism of $\mathbb{C}[X]$ which is not a morphism of $\mathbb{C}$-algebras.