Let $L$ be as before.

1. (1)

   There is a bijection between the set of approachable adelic toric metric on $L$ and the set of families of continuous concave functions $\{\psi_{v}\}_{v}$ on $N_{\mathbb{R}}$ such that $|\psi_{v}-\Psi|$ is bounded and $\psi_{v}=\Psi$ for all but finitely many $v$.

2. (2)

   There is a bijection between the set of approachable adelic toric metric on $L$ and the set of families of continuous concave functions $\{\psi^{\vee}_{v}\}_{v}$ on $\Delta_{\Psi}$ such that $\psi^{\vee}_{v}=0$ for all but finitely many $v$.