is thus also relatively compact in $C(\operatorname{\mathrm{Cham}}\Delta, \operatorname{\mathrm{Cham}}\Delta)$, hence $S$ is equicontinuous on $\operatorname{\mathrm{Cham}}\Delta$ by Arzela-Ascoli. Since $\operatorname{\mathrm{Auttop}}(\Delta)$ is a topological group, the inverse function of $\operatorname{\mathrm{Auttop}}(\Delta)$ is a homeomorphism and $S^{-1}$ is in turn relatively compact and equicontinuous on $\operatorname{\mathrm{Cham}}\Delta$.

Conversely, assume that $S$ and $S^{-1}$ are equicontinuous. We want to show that $S$ is relatively compact in $\operatorname{\mathrm{Auttop}}(\Delta)$. We already know by Arzela-Ascoli that $S$ is relatively compact in $C(\operatorname{\mathrm{Cham}}\Delta, \operatorname{\mathrm{Cham}}\Delta)$. A way to conclude is therefore to prove that the closure of $S$ in $C(\operatorname{\mathrm{Cham}}\Delta, \operatorname{\mathrm{Cham}}\Delta)$ is contained in $\operatorname{\mathrm{Auttop}}(\Delta)$. We thus consider a sequence $\varphi_n \to \varphi$ with $\{\varphi_n\} \subseteq \operatorname{\mathrm{Auttop}}(\Delta)$ and $\varphi \in C(\operatorname{\mathrm{Cham}}\Delta, \operatorname{\mathrm{Cham}}\Delta)$ and prove that $\varphi \in \operatorname{\mathrm{Auttop}}(\Delta)$. As a direct consequence of the fact that the adjacency relation is closed in a compact building, $\varphi$ is a (continuous) building morphism. To complete the proof, it suffices to find another such morphism $\psi$ so that $\varphi \psi = \psi \varphi = \operatorname{\mathrm{id}}_\Delta$. Since $S^{-1}$ is equicontinuous and thus relatively compact in $C(\operatorname{\mathrm{Cham}}\Delta, \operatorname{\mathrm{Cham}}\Delta)$, it follows that some subsequence of $(\varphi_n^{-1})$ converges to an element of $C(\operatorname{\mathrm{Cham}}\Delta, \operatorname{\mathrm{Cham}}\Delta)$. Taking this limit for $\psi$, we obtain $\varphi \psi = \psi \varphi = \operatorname{\mathrm{id}}_\Delta$.

To 