We assume that there are a normed finite-dimensional vector space $(V,\|\raisebox{1.72218pt}{.}\|)$ and a surjective homomorphism $V\otimes_{k}\mathscr{O}_{X}\to L$ such that $h$ is given by $\left\{|\raisebox{1.72218pt}{.}|^{\mathrm{quot}}_{(V,\|\raisebox{1.20552pt}{.}\|)}(x)\right\}_{x\in X^{\mathrm{an}}}$. Let $k^{\prime}$ be an extension field of $k$, and let $|\raisebox{1.72218pt}{.}|^{\prime}$ be a complete absolute value of $k^{\prime}$ as an extension of $|\raisebox{1.72218pt}{.}|$. We set

$$X^{\prime}:=X\times_{\operatorname{Spec}(k)}\operatorname{Spec}(k^{\prime}),\quad L=L\otimes_{k}k^{\prime}\quad\text{and}\quad V^{\prime}:=V\otimes_{k}k^{\prime}.$$

Let $\|\raisebox{1.72218pt}{.}\|^{\prime}$ be a norm of $V^{\prime}$ obtained by the scalar extension of $\|\raisebox{1.72218pt}{.}\|$. Moreover, let $h^{\prime}$ be a continuous metric of ${L^{\prime}}^{\mathrm{an}}$ given by the scalar extension of $h$. Then $h^{\prime}$ coincides with $\left\{|\raisebox{1.72218pt}{.}|^{\mathrm{quot}}_{(V^{\prime},\|\raisebox{1.20552pt}{.}\|^{\prime})}(x^{\prime})\right\}_{x^{\prime}\in{X^{\prime}}^{\mathrm{an}}}$.