From a Dessin d'Enfant on a surfaces $X$ we get a meromorphic function $f:X\rightarrow\mathbb{CP}^{1}$ such that the only critical values are $\{0,1,\infty\}$ i.e a Belyi pair $(X,f)$.
Belyi's theorem says that from this we get a Belyi pair $(X',f')$ where $X'$ and $f'$ are in fact defined over $\overline{\mathbb{Q}}$ and isomorphic to $(X,f)$ over $\mathbb{C}$.
Is this pair the unique (up to composing $f$ with automorphisms of $X$) pair over $\overline{\mathbb{Q}}$? I've seen this described in various places but want to know why this is true.
I'm trying to understand how $Gal(\overline{\mathbb{Q}},\mathbb{Q})$ acts on Dessin and don't have enough basic algebraic geometry to see that there are not choices on $(X',f')$ that could make this undefined.