Let $\Sigma$ be a complete fan. Then the map $\|\cdot\|\mapsto\psi_{\|\cdot\|}$ is a bijection between the space of integrable toric metrics on $\mathcal{O}(D_{\Psi})^{{\text{\rm an}}}$ and the space of functions $\psi\in{\overline{\mathscr{D}}}(N_{\mathbb{R}})$ such that $\operatorname{rec}(\psi)=\Psi$, were ${\overline{\mathscr{D}}}(N_{\mathbb{R}})$ is the space of functions of Definition 3.82.