group $\emptyset$-definable in $M$, and $G^*$ its interpretation in ${{\mathfrak C}}$. Group multiplication, denoted by $\cdot$, will be often skipped for simplicity, but sometimes we will write it explicitly.\
As above, a [*universal \[externally\] definable $G$-ambit*]{} is defined as an \[externally\] definable $G$-ambit which maps homomorphically (by a (unique) homomorphism of $G$-ambits) to an arbitrary \[externally\] definable $G$-ambit. It is clear (by general category theory reasons) that, in each of these two categories, if a universal $G$-ambit exists, then it is unique up to isomorphism. As mentioned in the introduction, $(G,S_{G,\operatorname{{ext}}}(M),\operatorname{{tp}}_{\operatorname{{ext}}}(e/M))$ is the unique up to isomorphism universal externally definable $G$-ambit, so, in the same way as above, we get a semigroup operation on $S_{G,\operatorname{{ext}}}(M)$. The existence of the universal definable $G$-ambit is justified in Remark \[remark: existence of universal definable G-ambit\]. The main point of the current paper is to describe the universal definable $G$-ambit as well as its local versions.

Let $\Delta$ be a subset of the set of all formulas (without parameters) in the language $\mathcal L$ in the variable $x$ of the same sort as $G$ and some parametric variables. By a [*$\Delta$-formula*]{} over $M$ we mean a Boolean combination of instances of formulas from $\Delta$ 