Let $L/k$ be a field extension of algebraically closed fields of characteristic zero. Let $U$ be a smooth quasi-projective variety over $k$.

I am trying to understand why the base-change functor from $k$ to $L$ from

- the category of finite etale $k$-morphisms to $U$

to

- the category of finite etale $L$-morphisms to $U_L$

is faithful.

I have the feeling it is almost "trivial" and that there is a much more general statement concerning the base-change functor.

Does this appear anywhere, or could anybody provide some explanation?