Assume the previous hypothesis.

1. (1)

   The assignment $\|\cdot\|\mapsto\psi_{\|\cdot\|}$ is a bijection between the space of approachable toric metrics on $L^{{\text{\rm an}}}$ and the space of continuous concave functions $\psi$ on $N_{\mathbb{R}}$ such that $|\psi-\Psi|$ is bounded.

2. (2)

   The assignment $\|\cdot\|\mapsto\psi_{\|\cdot\|}^{\vee}$ is a bijection between the space of approachable toric metrics on $L^{{\text{\rm an}}}$ and the space of continuous concave functions on $\Delta_{\Psi}$.