The map $\tau\colon\overline{{\mathbb{D}}}_{r}\to{\mathcal{M}}(A_{r})$ given by (A.1) is clearly well defined. It is also continuous on $\overline{{\mathbb{D}}}^{*}_{r}$. To prove continuity at $0$, we note that for each $f\in A_{r}$, we can write $f^{\operatorname{hol}}=z^{\operatorname{ord}_{0}(f)}u$, where $u$ is a continuous function on $\overline{{\mathbb{D}}}_{r}$ that is holomorphic on ${\mathbb{D}}_{r}$ with $u(0)\neq 0$. As a consequence, we get $\lim_{z\to 0}\frac{\log|f^{\operatorname{hol}}(z)|_{\infty}}{\log|z|_{\infty}}=\operatorname{ord}_{0}(f)$.

Now, for each $\rho\in(0,1]$, $\lambda^{-1}(\rho)\subset C_{\mathrm{hyb}}(r)$ can be identified with the circle of radius $r$ with respect to the absolute value $|\cdot|_{\infty}^{\rho}$, while $\lambda^{-1}(0)$ is the non-Archimedean absolute value $r^{-\operatorname{ord}_{0}}$ on ${\mathbb{C}}(\!({t})\!)$. This proves that the map $\tau$ above is bijective, and hence a homeomorphism by compactness. ∎