Let $({\mathcal{X}}_{2},{\mathcal{L}}_{2},e)$ be a proper model of $(X_{2},L_{2})$ which induces the metric in ${\overline{L}}_{2}$, and ${\mathcal{X}}^{\prime}_{1}$ be a proper model of $X_{1}$. Let ${\mathcal{X}}_{1}$ be the Zariski closure of the graph of $\varphi$ in ${\mathcal{X}}_{1}^{\prime}\times_{S}{\mathcal{X}}_{2}$. This is a proper model of $X_{1}$ equipped with a morphism $\varphi_{S}\colon{\mathcal{X}}_{1}\to{\mathcal{X}}_{2}$. Then $({\mathcal{X}}_{1},\varphi_{S}^{*}{\mathcal{L}}_{2},e)$ is a proper model of $(X_{1},\varphi^{*}L_{2})$ which induces the metric of $\varphi^{*}{\overline{L}}_{2}$. ∎