$*$ on $S_G(M)/E_\Delta$ which is given by $$p/E_\Delta * q/E_\Delta = \lim_{g \rightarrow p/E_\Delta} g(q/E_\Delta),\label{eq: 0}$$ equivalently, $$a/E_\Delta' * b/E_\Delta'= \lim_{g \rightarrow a/E_\Delta'} g(a/E_\Delta').$$ (When we write “$g$ tends to $p/E_\Delta$”, we mean “$\operatorname{{tp}}_\Delta(g/M)/E_\Delta$ tends to $p/E_\Delta$ with $g$ ranging over $G$”, i.e. $\lim_{g \rightarrow p/E_\Delta} g(q/E_\Delta)=r/E_\Delta$ if and only if for every open neighborhood $U$ of $r/E_\Delta$ there is an open neighborhood $V$ of $p/E_\Delta$ such that for all $g \in G$ such that $\operatorname{{tp}}_\Delta(g/M)/E_\Delta \in V$ one has $g(q/E_\Delta) \in U$. And analogously in the equivalent definition.)

To simply notation, denote $(G,S_{G,\Delta}(M)/E_\Delta,\operatorname{{tp}}_{\Delta}(e/M)/E_\Delta)$ by $(G,\mathcal U,x_0)$. By the universality of $(G,{\mathcal U},x_0)$, for every $x \in {\mathcal U}$ there is a unique $f_x \colon {\mathcal U} \to {\mathcal U}$ which is a $G$-flow homomorphism mapping $x_0$ to $x$. For $p \in {\mathcal U}$ put $$p * x = f_x(p).$$ From the choice of $f_x$, the following properties follow immediately.

1.  $*$ is continuous in the left coordinate.

2.  $*$ extends the action of $G$, i.e. $(gx_0) * x = gx$ for all $g \in G$ and $x \in {\mathcal U}$.

3.  $\lim_{g \rightarrow p} gq = p * q$ for all $p,q \in {\mathcal U}$ (here $g \rightarrow p$ 