We always consider the $G$-topology induced by the strictly $K$-affinoid domains. We claim that the map $q\colon(X^{\prime})^{\rm an}\to{X^{{\mathrm{an}}}}$ is a homeomorphism and that it also identifies the $G$-topologies. In fact, this follows easily from the following claim:

Step 1: Let $V$ be a strictly affinoid space over $K$ and $V^{\prime}\coloneqq V\hat{\otimes}_{K}K^{\prime}$. Then the natural projection $q\colon V^{\prime}\to V$ is a homeomorphism which identifies the $G$-topologies.

Let $p^{e}=[K^{\prime}:K]$ be the degree of the purely inseparable field extension. It is clear that for every $g\in\mathcal{O}(V^{\prime})$, there is $f\in\mathcal{O}(V)$ with

This property easily shows that $q\colon V^{\prime}\to V$ is a homeomorphism which we read now as an identification. Using that (2.6) holds also for rational functions $g$ on $V^{\prime}$ and $f$ on $V$, we see that $V$ and $V^{\prime}$ have the same strictly rational domains. By the Gerritzen–Grauert theorem [BGR84, Cor. 7.3.5/3], we deduce the Step 1.

Next we prove the bijective correspondence between the model metrics on $L$ and on $L^{\prime}$. For this, it is enough to show that we have a bijective correspondence between model functions on ${X^{{\mathrm{an}}}}$ and model functions on $(X^{\prime})^{\rm an}$.

We recall from [GM16, Def. 2.8, 2.11] that a piecewise $\mathbb{Q}$-linear function on a strictly $K$-analytic space $W$ is a function $f:W\to\mathbb{R}$ such that there is a $G$-covering $\{U_{i}\}_{i\in I}$ of $W$ by strictly affinoid domains, analytic functions $\gamma_{i}\in\mathcal{O}(U_{i})^{\times}$ and non-zero $m_{i}\in\mathbb{N}$ with $m_{i}f=-\log|\gamma_{i}|$ on $U_{i}$ for every $i\in I$.

By [GM16, Rem. 2.6, Prop. 2.10], model functions and piecewise $\mathbb{Q}$-linear functions are the same and hence we have to check the bijective correspondence between piecewise $\mathbb{Q}$-linear functions on ${X^{{\mathrm{an}}}}$ and $(X^{\prime})^{\rm an}$. This can be checked $G$-locally and hence it is enough to prove the following:

Step 2: Using the same assumptions as in Step 1, the map $f\mapsto f\circ q$ is an isomorphism from the group of piecewise $\mathbb{Q}$-linear functions on $V$ onto the group of piecewise $\mathbb{Q}$-linear functions on $V^{\prime}$.

Using the above definition of piecewise $\mathbb{Q}$-linear functions, Step 1 and (2.6) yield easily Step 2.

To deduce the lemma, it remains to check that the identification between the model metrics on $L$ and $L^{\prime}$ preserves semipositivity. This is an easy consequence of the projection formula applied to finite morphisms between closed curves in the special fibers of models. ∎