Let $\Sigma$ be a complete fan in $N_{\mathbb{R}}$ and $\Psi$ a virtual support function on $\Sigma$. Then the correspondence $(\Pi,\psi)\mapsto({\mathcal{X}}_{\Pi},D_{\psi})$ is a bijection between:

• $\bullet$

  the set of pairs $(\Pi,\psi)$, where $\Pi$ is a complete SCR polyhedral complex in $N_{\mathbb{R}}$ with $\operatorname{rec}(\Pi)$= $\Sigma$ and $\psi$ is an H-lattice function on $\Pi$ such that $\operatorname{rec}(\psi)=\Psi$;

• $\bullet$

  the set of isomorphism classes of toric models $({\mathcal{X}},D)$ of $(X_{\Sigma},D_{\Psi})$.