isomorphism of divisorial exact sequences $$\xymatrix{0\ar[r]& M \ar[d]^-{\operatorname{id}_M}_-{I_n}\ar[r]^-{div}_-{V^T} & \Weil(X)\ar[d]^-{\vf^*}_-{I_m}\ar[r]^-d_-Q & \operatorname{Cl}(X)\ar[d]^-{\overline{\vf}^*}_-{I_r} \ar[r] & 0\\
     0\ar[r]& M \ar[r]^-{div}_-{V^T} & \Weil(Y)\ar[r]^-d_-Q & \operatorname{Cl}(Y) \ar[r] & 0}$$ where $\vf^*$ is the pull-back of Weil divisors and $\overline{\vf}^*$ is the induced morphism on divisor classes. By composing $\overline{\vf}^*$ with isomorphisms $\operatorname{Pic}(X)_\R\cong\R^r$ and $\operatorname{Pic}(Y)_\R\cong\R^r$, defined in the previous Proposition \[prop:nef\], there follows the following identification of nested divisorial sub-cones of the positive orthant $\R^r_+$ $$\xymatrix{\operatorname{Nef}(\I_{\Si})\cong\operatorname{Nef}(X)\ar@{^(->}[dd]\ar@<-1ex>@{^(->}[rd]&\\
     &\left(\operatorname{Mov}(Q)\cong\overline{\operatorname{Mov}}(X)\stackrel{\overline{\vf}^*_\R}{\cong}\overline{\operatorname{Mov}}(Y)\right)
     \,\subseteq\, \left(\langle Q\rangle\cong\overline{\operatorname{Eff}}(X)\stackrel{\overline{\vf}^*_\R}{\cong}\overline{\operatorname{Eff}}(Y)\right)\\
     \operatorname{Nef}(\I_{\Xi})\cong\operatorname{Nef}(Y)\ar@{^(->}[ur]&
}$$ Moreover, for every Weil divisor $D=\sum_{\rho\in\Si(1)}a_\rho D_\rho\in\Weil(X)$, there follows the identification of associated polytopes $$\{\m\in M_\R\,|\,V^T\cdot\m \geq -\aa\}=\D_D=\operatorname{id}_M(\D_D)=\D_{\vf^*(D)}$$ where $\aa=(a_\rho)_{\rho\in\Si(1)}$. In particular $D$ is semi-ample if and only if its class $[D]$ belongs to $\operatorname{Nef}(X)$. Then there exists a positive integer $k\in\N$ such that $$\operatorname{Conv}(\{\m_\s\in M\,|\,\s\in\Si(n)\})=k\D_D\stackrel{\operatorname{id}_M}{=}k\D_{\vf^*(D)}
     =\operatorname{Conv}(\{\m_I\in M\,|\,I\in\I_{\Xi(n)}\})$$ where $$\label{mI}
\m_I:=-k(V_I^T)^{-1}\cdot \aa_I$$ being $\aa_I$ the subvector of $\aa$ whose entries are indexed by $I$. In particular, this means that, for every $\s\in\Si(n)$, $\m_\s=\m_I$ for any $I\in\I_{\Xi(n)}$ such that $\langle V_I\rangle\subseteq \s$.

Results on divisorial cones are direct consequences of the previous Proposition \[prop:nef\]. In particular[^1] $$\operatorname{Nef}(\I_\Si)=\bigcap_{I\in\I_\Si} \langle Q^I\rangle\subseteq\bigcap_{J\in\I_\Xi} \langle Q^J\rangle = \operatorname{Nef}(\I_\Xi)$$ because $\Xi$ is a refinement of $\Si$, so implying that $$\forall\,I\in\I_\Si\ \exists\,J\in\I_\Xi\,: \langle V_J\rangle \subseteq \langle V_I\rangle\ \Longleftrightarrow\ \langle Q^I\rangle \subseteq 