Surjectivity follows from Proposition 4.3, and continuity from (4.2) after unwinding the definitions. It remains to establish (4.2). Consider any point $\xi^{\prime}\in{\mathcal{X}}_{0}$ and set $\xi=\pi(\xi^{\prime})$. We can find adapted coordinate charts $({\mathcal{U}}^{\prime},z^{\prime})$ at $\xi^{\prime}$ on ${\mathcal{X}}^{\prime}$ and $({\mathcal{U}},z)$ at $\xi$ on ${\mathcal{X}}$ such that $\rho({\mathcal{U}}^{\prime})\subset{\mathcal{U}}$ and such that the following holds: $t=\prod_{i=0}^{p}z_{i}^{b_{i}}$ in ${\mathcal{U}}$, $t=\prod_{j=0}^{p^{\prime}}(z^{\prime}_{j})^{b^{\prime}_{j}}$ in ${\mathcal{U}}^{\prime}$ and $\rho^{*}z_{i}=\prod_{j}(z^{\prime}_{j})^{a_{ij}}$. Since the map $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}$ is given by (4.1), the result now follows from Proposition 2.1. ∎