Let $({\mathcal{X}},D,e)$ be a semipositive toric model. By Theorem 4.81, to the pair $({\mathcal{X}},D)$ corresponds a pair $(\Pi,\psi^{\prime})$, where $\psi^{\prime}$ is an H-lattice function on $\Pi$, $\operatorname{rec}(\Pi)=\Sigma$ and $\operatorname{rec}(\psi^{\prime})=e\Psi$. By Theorem 4.95, the function $\psi^{\prime}$ is concave. We put $\psi=\frac{1}{e}\psi^{\prime}$. It is clear that equivalent models produce the same function.

Conversely, let $\psi$ be a rational piecewise affine concave function. Let $\Pi^{\prime}=\Pi(\psi)$. This is a rational polyhedral complex. Let $\Sigma^{\prime}=\operatorname{rec}(\Pi^{\prime})$. This is a conic rational polyhedral complex. By Proposition 3.72, $\Sigma^{\prime}=\Pi(\Psi)$. Since $\Psi$ is a support function on $\Sigma$, we deduce that $\Sigma$ is a refinement of $\Sigma^{\prime}$. Put $\Pi=\Pi^{\prime}\cdot\Sigma$ (Definition 3.10). Since $\Pi^{\prime}$ is a rational polyhedral complex and $\Sigma$ is a fan, then $\Pi$ is an SCR polyhedral complex. Moreover, by Lemma 3.11, we have

Let $e>0$ be an integer such that $e\psi$ is an H-lattice function. Then $({\mathcal{X}}_{\Pi},D_{e\psi},e)$ is a toric model of $(X_{\Sigma},D_{\Psi})$. Both procedures are inverse of each other. ∎