A morphism $f:X\to Y$ of schemes is said to be surjective if its underlying map of sets is surjective. Tag 0487, Tag 025Z, and Tag 025T imply that $f$ is surjective if and only if the induced map $$ f_\ast:X(k)\to Y(k) $$ is surjective, for every algebraically closed field $k$.
Now, consider the canonical morphism of schemes $f:\mathop{\mathbf{Spec}}\mathbb{C}\to \mathop{\mathbf{Spec}}\mathbb{Q}$, induced by the field extension of the complex numbers over the rational numbers. Obviously, the morphism $f$ is surjective. However, the canonical morphism $\mathop{\mathbf{Spec}}\overline{\mathbb{Q}}\to \mathop{\mathbf{Spec}}\mathbb{Q}$ does not lift to $\mathop{\mathbf{Spec}}\mathbb{C}$, where $\overline{\mathbb{Q}}$ is an algebraic closure of $\mathbb{Q}$, i.e. $f_\ast$ is not surjective for the algebraically closed field $\overline{\mathbb{Q}}$.
Could you please help identify what is wrong?