ScalingStacks

Proof. By Proposition 3.20, the left hand side $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi))$ is identified with $p(\boldsymbol{0})^{\mathrm{an}}(\overline{\mathbb{D}}^{\vee}(L,\phi))$, which is contained in the open subset $p(\boldsymbol{0})^{\mathrm{an}}(\mathbb{D}^{\vee}(L,\phi(\epsilon)))$. This open subset is contained in $p(\boldsymbol{0})^{\mathrm{an}}(\overline{\mathbb{D}}^{\vee}(L,\phi(\epsilon)))$ which is identified with $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi(\epsilon)))$, hence this open subset is contained in the topological interior of the later, the right hand side. ∎