It follows from (4.4) and (4.5) that $\tau$ is a bijection. Since $X_{A_{r}}^{\mathrm{An}}$ is compact and $X_{\overline{{\mathbb{D}}}_{r}}^{\mathrm{hyb}}$ is Hausdorff, it only remains to prove that $\tau$ is continuous. It suffices to show that the corresponding map $\tau_{\mathcal{X}}\colon X_{A_{r}}^{\mathrm{An}}\to{\mathcal{X}}^{\mathrm{hyb}}_{\overline{{\mathbb{D}}}_{r}}$ is continuous for a given snc model ${\mathcal{X}}$. For this, in turn, it suffices to show that $\operatorname{Log}_{\mathcal{X}}\circ\tau_{\mathcal{X}}$ is continuous near the central fiber.

Consider a coordinate chart $({\mathcal{U}},z)$ adapted to ${\mathcal{X}}_{0}$ in the sense of §2.2. Let $E_{0},\dots,E_{p}$ be the irreducible components of ${\mathcal{X}}_{0}$ intersecting ${\mathcal{U}}$. Let $\hat{\mathcal{U}}\subset X_{A_{r}}^{\mathrm{An}}$ be the set of seminorms satisfying $|z_{i}|<1$ for $0\leq i\leq p$. Then we have

on $\hat{\mathcal{U}}\setminus\pi^{-1}(0)$. Now the function $(\log|z_{i}|^{-1})_{i}$ is continuous on $\hat{\mathcal{U}}$ with values in the simplex $\sigma={\mathbb{R}}_{+}^{p+1}\cap\{\sum_{0}^{p}b_{i}w_{i}=1\}\subset\Delta({\mathcal{X}})$. This completes the proof, since we can cover a neighborhood of the central fiber in $X_{A_{r}}^{\mathrm{An}}$ with sets of the type $\hat{{\mathcal{U}}}$. ∎