Denote by $\iota\colon X_{\Sigma}=X_{\operatorname{rec}(\Pi)}\to{\mathcal{X}}_{\Pi}$ the open immersion of the generic fibre. The recession function (Definition 3.85) determines the restriction of the $\mathbb{T}$-Cartier divisor to the fibre over the generic point. Therefore, when $\psi$ is an H-lattice function on $\Pi$ with $\operatorname{rec}(\psi)=\Psi$, we have that

(4.82)

$$\iota^{\ast}D_{\psi}=D_{\operatorname{rec}(\psi)}=D_{\Psi}.$$

Thus $({\mathcal{X}}_{\Pi},D_{\psi})$ is a toric model of $(X_{\Sigma},D_{\Psi})$. The statement follows from Theorem 4.60 and Theorem 4.80. ∎