Let $k$ be an algebraically closed field. By a finitely generated subfield of $k$ I mean a subfield $k_0\subset k$ that is finitely generated over the prime subfield of $k$ (that is, over $\mathbb Q$ or $\mathbb F_p$).

I need a reference or a proof for the following (well-known? evident?) proposition:

Proposition.Let $$f\colon X\to V$$ be a morphism of $k$-varieties. Then the triple $(X,Y,f)$ can be defined over a finitely generated subfield of $k$. In other words, there exists a finitely generated subfield $k_0\subset k$ and a morphism of $k_0$-varieties $$f_0\colon X_0\to Y_0$$ such that $(X_0,Y_0,f_0)\times_{k_0} k$ is isomorphic to $(X,Y,f)$.

A referee asked me to prove this assertion. I know how to prove this for affine and projective varieties, but not for *abstract* varieties.