Consider the category $\mathbf{C}$ of groups with a given operator domain $\Omega$. It's true that for every set $X$ there is a free object $G$ on $X$?
I believe it's true and that the underlying group to $G$ must be a free group on $X$. But I don't know how to define the action of the operators in such a way that $G$ becomes a free object.