Let $\Sigma$ be a complete fan and let $\Psi$ be a support function on $\Sigma$. Let $\psi$ be a rational piecewise affine concave function on $N_{\mathbb{R}}$ with $\operatorname{rec}(\psi)=\Psi$ and let $\|\cdot\|$ be the metric defined by any model of $(X_{\Sigma},L_{\Psi})$ in the equivalence class determined by $\psi$. Then the equality $\psi_{\|\cdot\|}=\psi$ holds. So the metric $\|\cdot\|$ agrees with the metric $\|\cdot\|_{\psi}$ of Proposition 5.16. Moreover, the algebraic metric $\|\cdot\|$ is semipositive.