I am a complete novice in the art of categorification, so this may not be a great question.

**Background**.

Question.Is there a categorification $\mathbf C$ of the free monoid on a finite set $X$ together with a ''forgetful" functor $\mathbf C\longrightarrow \mathbf {FSet}/X$ categorifying the abelianization map such that $\mathbf C^{\mathbf C}$ categorifies generating functions in non-commuting variables $X$.

I assume the answer will be some sort of "free" monoidal category. Please take into account that I am a category-friendly mathematician, but not a category theorist, when formulating your answer.

**Vague motivation.** Is there a categorification of the
Chomsky-Schützenberger theorem on context-free grammars and algebraic power series?

I realize that species categorify exponential generating fuctions and Chomsky-Schützenberger is about ordinary generating functions, but this shouldn't matter.

**Update.**. The free monoidal category is no good because it only has identity morphisms. It doesn't even work to recover the 1-generated case.