Suppose that $S$ is a complex projective surface defined over $\overline{\mathbb Q}$, namely there exists a surface $S_{\overline{\mathbb Q}}$ over $\overline{\mathbb Q}$ such that: $$S_{\overline{\mathbb Q}}\times_{\operatorname{Spec} \overline{\mathbb Q}}\operatorname{Spec} \mathbb C$$

Now consider a $(-1)$-curve $E\hookrightarrow S$ (namely $E\cong\mathbb P^1_{\mathbb C}$ and $E^2=-1$), and suppose that $f:S\longrightarrow S'$ is the blow-down (contraction) of $E$ on a point, where $S'$ is another surface.

I don't understand the following statement:

There exists a surface $S'_{\overline{\mathbb Q}}$ over $\overline{\mathbb Q}$ and a morphism $$ g : S_{\overline{\mathbb Q}} \longrightarrow S'_{\overline{\mathbb Q}} $$

such that $f = g \times_{\operatorname{Spec}\overline{\mathbb Q}}\mathrm{id}_{\operatorname{Spec}\mathbb C}$. In other words the blow-down $f$ is a morphism defined over $\overline{\mathbb Q}$. (See this question for the field of definition of a morphism.)

The thesis should follows from the fact that, by definition, every $(-1)$-curve is defined over $\overline{\mathbb Q}$.

Many thanks in advance.