Let $\Lambda_{i}\in\Pi_{i}$ such that $A(\Lambda_{1})\subset\Lambda_{2}$. Then the map ${\widetilde{M}}_{2}\to{\widetilde{M}}_{1}$ given by $(m,l)\mapsto(H^{\vee}m,\langle{\operatorname{val}}(p),m\rangle+l)$ for $m\in M$ and $l\in\mathbb{Z}$ (which is just the dual of the linearization of $A$) induces a morphism of semigroups ${\widetilde{M}}_{2,\Lambda_{2}}\to{\widetilde{M}}_{1,\Lambda_{1}}$. Since $\chi^{m}(p)\varpi^{-\langle{\operatorname{val}}(p),m\rangle}$ belongs to $K^{\circ}$, the assignment

$$\chi^{(m,l)}\longmapsto(\chi^{m}(p)\varpi^{-\langle{\operatorname{val}}(p),m\rangle})\chi^{(H^{\vee}m,\langle{\operatorname{val}}(p),m\rangle+l)}$$

defines a ring morphism $K^{\circ}[{\widetilde{M}}_{2,\Lambda_{2}}]\to K^{\circ}[{\widetilde{M}}_{1,\Lambda_{1}}]$. This morphism sends $\chi^{(0,1)}-\varpi$ to $\chi^{(0,1)}-\varpi$, hence induces a morphism $K^{\circ}[{\mathcal{X}}_{\Lambda_{2}}]\to K^{\circ}[{\mathcal{X}}_{\Lambda_{1}}]$ and a map ${\mathcal{X}}_{\Lambda_{1}}\to{\mathcal{X}}_{\Lambda_{2}}$. Varying $\Lambda_{1}$ and $\Lambda_{2}$ we obtain maps, that glue together into a map

$$\Phi_{p,A}\colon{\mathcal{X}}_{\Pi_{1}}\longrightarrow{\mathcal{X}}_{\Pi_{2}}.$$

By construction, this map extends $\varphi_{p,H}$ and is equivariant with respect to the morphism $\rho_{H}$. ∎