The map of topological spaces is given in Proposition 2.88. For the ring homomorphism, it suffices to construct a $k$-algebra homomorphism $\mathscr{O}_{Z^{an}}(U)\to\mathcal{A}_{V}$ for any open set $U\subseteq\mathfrak{M}(\mathcal{A})$ and any affinoid domain $V\subseteq U$. Moreover, it suffices to consider $U$ and $V$ of basic form

$$U=U(\underline{p}^{-1}\underline{f},\underline{q}\underline{g}^{-1}),\quad V=\mathfrak{M}(\mathcal{A}_{Z}((\underline{p}-\underline{\epsilon})^{-1}\underline{f},(\underline{q}+\underline{\epsilon})\underline{g}^{-1}))\text{ , }\epsilon>0$$

There is a homomorphism of $k$-algebras $\mathcal{R}^{\mathrm{an}}(U)\to\mathcal{A}_{Z}((\underline{p}-\underline{\epsilon})^{-1}\underline{f},(\underline{q}+\underline{\epsilon})\underline{g}^{-1})$ sending $\frac{\tilde{f}}{\tilde{g}}$ for $\tilde{f},\tilde{g}\in A_{Z}$ to itself, the later being an element of $\mathcal{A}_{V}$ since $\frac{1}{\tilde{g}}\in\mathcal{A}_{V}$ by Lemma 2.25. As uniform limits of sequence in $\mathcal{R}^{\mathrm{an}}(U)$ remains to be uniform limits, this homomorphism extends to a $k$-algebra homomorphism $\mathscr{O}_{Z^{an}}(U)\to\mathcal{A}_{V}$. ∎