Let $(\|\cdot\|_{v})_{v\in M_{\mathbb{K}}}$ be a metric on $L$ and write ${\overline{L}}=(L,(\|\cdot\|_{v})_{v\in M_{\mathbb{K}}})$. Suppose first that ${\overline{L}}$ is toric and quasi-algebraic. Let $S\subset M_{\mathbb{K}}$ be a finite set containing the Archimedean places, $\mathbb{K}^{\circ}_{S}$ as in Definition 2.51, $e\geq 1$ an integer and $({\mathcal{X}},{\mathcal{L}})$ a proper model over $\mathbb{K}^{\circ}_{S}$ of $(X_{\Sigma},L^{\otimes e})$ so that $\|\cdot\|_{v}$ is induced by the localization ${\mathcal{L}}_{v}$ for all $v\notin S$. Over $\mathbb{K}$, there is an isomorphism from $({\mathcal{X}},{\mathcal{L}})$ to the canonical model $({\mathcal{X}}_{\Sigma},{\mathcal{L}}_{e\Psi})$. Since $\mathbb{K}^{\circ}_{S}$ is Noetherian, this isomorphism and its inverse are defined over $\mathbb{K}^{\circ}_{S^{\prime}}$ for certain finite subset $S^{\prime}$ containing $S$. Thus, enlarging the finite set $S$ if necessary, we can suppose without loss of generality that $({\mathcal{X}},{\mathcal{L}})$ agrees with the canonical model $({\mathcal{X}}_{\Sigma},{\mathcal{L}}_{e\Psi})$. Hence, $\|\cdot\|_{v}=\|\cdot\|_{v,e\Psi}^{1/e}=\|\cdot\|_{v,\Psi}$ for all places $v\notin S$. In consequence, it is an adelic toric metric.

Conversely, suppose that ${\overline{L}}$ is a toric adelic metrized line bundle. Let $S$ be the union of the set of Archimedean places and $\{v\in M_{\mathbb{K}}|\psi_{v}\neq\Psi\}$. By definition, this is a finite set. Let $({\mathcal{X}}_{\Sigma},{\mathcal{L}}_{\Psi})$ be the canonical model over $\mathbb{K}^{\circ}_{S}$ of $(X_{\Sigma},L)$. Then $\|\cdot\|_{v}$ is the metric induced by this model, for all $v\notin S$. Hence ${\overline{L}}$ is quasi-algebraic. ∎