If I have some free monoid: $A^\ast$, generated from $A$.
I can define a function that undoes any application (concatenation), of a element of $A$ to an element of $A^\ast$: $$f:\; A^\ast \to A^\ast$$
It is given by: $\forall g\in A$, for $\forall a \in A^\ast$, and for $\epsilon$ the identity element:
$$ag \mapsto a$$ $$\epsilon \mapsto \epsilon$$
This covers all elements of $A^\ast$ because every non-identity element of a free monoid can be expressed as another element, with one of the generators applied to it. It is a function (though showing that is a little harder)
Note that since $\epsilon g = g$ $$g \mapsto \epsilon$$
Is there a name for $f$?
I have been thinking of it as the parent function