Symmetry and reflexivity are straightforward. For transitivity assume that we have toric models $({\mathcal{X}}_{i},D_{i},e_{i})$, $i=1,2,3$, that the first and second model are equivalent through $({\mathcal{X}}^{\prime},D^{\prime},e^{\prime})$ and that the second and the third are equivalent through $({\mathcal{X}}^{\prime\prime},D^{\prime\prime},e^{\prime\prime})$. Then, by Theorem 4.60, ${\mathcal{X}}^{\prime}$ and ${\mathcal{X}}^{\prime\prime}$ are defined by SCR polyhedral complexes $\Pi^{\prime}$ and $\Pi^{\prime\prime}$ respectively, with $\operatorname{rec}(\Pi^{\prime})=\operatorname{rec}(\Pi^{\prime\prime})=\Sigma$. Let $\Pi^{\prime\prime\prime}=\Pi^{\prime}\cdot\Pi^{\prime\prime}$. By Lemma 3.11, $\operatorname{rec}(\Pi^{\prime\prime\prime})=\Sigma$. Thus $\Pi^{\prime\prime\prime}$ determines a model ${\mathcal{X}}^{\prime\prime\prime}$ of $X_{\Sigma}$. This model has morphisms $\beta^{\prime}$ and $\beta^{\prime\prime}$ to ${\mathcal{X}}^{\prime}$ and ${\mathcal{X}}^{\prime\prime}$ respectively. We put $e^{\prime\prime\prime}=e^{\prime}e^{\prime\prime}$ and $D^{\prime\prime\prime}=e^{\prime\prime}\beta^{\prime}{}^{\ast}D^{\prime}=e^{\prime}\beta^{\prime\prime}{}^{\ast}D^{\prime\prime}$. Now it is easy to verify that $({\mathcal{X}}^{\prime\prime\prime},D^{\prime\prime\prime},e^{\prime\prime\prime})$ provides the transitivity property. ∎