Let $\Sigma$ be a complete fan in $N_{\mathbb{R}}$. Let $\Psi$ be a support function on $\Sigma$. Then the correspondence of Theorem 4.81 induces a bijective correspondence between the set of rational piecewise affine concave functions $\psi$ with $\operatorname{rec}(\psi)=\Psi$ and the set of equivalence classes of semipositive toric models of $(X_{\Sigma},D_{\Psi})$ over $S$.