\to S_{G,\Delta}(M)/F_1$. We see that $f'$ is a homomorphism from the $G$-ambit $(G,S_{G,\Delta}(M), \operatorname{{tp}}_{\Delta}(e/M))$ to $(G,S_{G,\Delta}(M)/F_2, \operatorname{{tp}}_{\Delta}(e/M)/F_2)$. But there is only one such homomorphism, and it is given by $p \mapsto p/F_2$. Therefore, $f(p/F_1)=p/F_2$ for any $p$, hence $F_1 \subseteq F_2$.\
ii) By assumption, $(G,S_G(M), \operatorname{{tp}}(e/M))$ is a definable $G$-ambit. Hence, Corollary \[cor: definable ambit is a quotient\] implies that $(G,S_G(M), \operatorname{{tp}}(e/M))$ is the universal definable $G$-ambit (literally, the quotient $S_G(M)/E$ is universal definable, but this implies the universal property of $S_G(M)$). And we finish using (i).

Define an equivalence relation $E_\Delta'$ on $G^*$ by $$E_\Delta'(a,b) \iff E_\Delta(\operatorname{{tp}}_{\Delta}(a/M),\operatorname{{tp}}_{\Delta}(b/M)).$$ In the definable context, we will skip $\Delta$ and write $E'$. We see that $E_\Delta'$ is a type-definable over $M$, bounded equivalence relation on $G^*$ which is coarser than the relation of having the same complete $\Delta$-type over $M$. We have a natural topological identification of $S_{G,\Delta}(M)/E_\Delta$ with $G^*/E_\Delta'$, which we will be using freely.

Our main goal is to give an explicit description of the relation $E_\Delta$, equivalently of $E_\Delta'$.

By Corollary \[cor: definable ambit is a quotient\], a standard argument (a sketch of which we give below for the reader’s convenience) shows that there is a left continuous semigroup operation 