As in the proof of Proposition 5.16, we can verify that the function $\psi_{\|\cdot\|}-\Psi$ can be extended to a continuous function on $N_{\Sigma}$. Using that $\theta_{0}$ is a section of $\rho_{0}$ and the image of $\theta_{0}$ consists of points which are fixed under the action of $\mathbb{S}^{{\text{\rm an}}}$, we also verify that $\|\cdot\|_{\mathbb{S}}$ is the toric metric associated to $\psi_{\|\cdot\|}$ by the same proposition. ∎