Let $\mathrm{X}$ be the analytic space associated to a proper $K$-scheme and let $f\colon\mathrm{X}\rightarrow\mathrm{X}$ be a finite morphism. Let $L$ be a line bundle on $\mathrm{X}$, $d$ an integer such that $d\geq 2$ and an isomorphism $\varepsilon\colon f^{*}L\simeq L^{d}$. The line bundle $L$ possesses a unique continuous metric such that the isomorphism $\varepsilon$ is an isometry. If $L$ is ample, then this metric is semi-positive.