Also in ME.

Let $k$ be a field complete with respect to a non-trivial non-archimedean

absolute value (so that rigid $k$-space makes sense). Denote $K$ a finite field extension of $k$.

Denote $Xrightsquigarrow X^{mathrm{an}/k}$ the analytification functor from the category of locally of finite type $k$-schemes to the category of rigid $k$-spaces.

Similarly there is an analytification functor $Xrightsquigarrow X^{mathrm{an}/K}$ over $K$.

There is a well-defind forgetful functor $S:Xrightsquigarrow X$ from $K$-schemes to $k$-schemes ($S$ represents schemes) and a forgetful functor $R:Yrightsquigarrow Y$ from rigid $K$-spaces to rigid $k$-spaces ($R$ represents rigid).

Let $X$ be a locally of finite type $K$-scheme. I believe that $S(X)^{mathrm{an}/k}cong R(X^{mathrm{an}/K})$ as rigid $k$-spaces. The universal property induces a canonical map $R(X^{mathrm{an}/K})to S(X)^{mathrm{an}/k}$ but I cannot show it is an isomorphism. A proof or reference would be nice.

p.s. the idea comes from proving absolute/relative Frobenius morphism commutes with analytification, but I first need to make sure the maps have the same source.