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