Let $(\mathcal{X},\mathcal{L},e)$ and $(\mathcal{X}^{\prime},{\mathcal{L}}^{\prime},e^{\prime})$ be proper models of $(X,L)$, and $f\colon\mathcal{X}^{\prime}\to\mathcal{X}$ a morphism of models such that $({\mathcal{L}}^{\prime})^{\otimes e}\simeq f^{\ast}{\mathcal{L}}^{\otimes e^{\prime}}$. Then the metrics on $L^{{\text{\rm an}}}$ induced by both models agree.