# Thompson's group F and monoidal categories

Fiore and Leinster have proved that if $\mathcal{A}$ is a free monoidal category generated by one object $A$ such that there exists an isomorphism $\alpha: A \otimes A \to A$, then for every object $X \in \mathcal{A}, Aut(X)$ is isomorphic to the Thompson group $F$.

My question is the following: if we assume instead that there exist a morphism $\alpha: A \otimes A \to A$ (not necessarily an isomorphism), and a morphism $\beta: A \to A \otimes A$ such that $\alpha \circ \beta = id$, is the result of Fiore and Leinster still true ? Or at least $F \subset Aut(X)$ ?

I have a feeling we at least have $F \subset Aut(X)$ by replacing the maps $\alpha^{-1}$ in Fiore's text by $\beta$ but this is as far as I went...

Edit : cross-post on MO

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This question might be (more) appropriate for MO. –  Rasmus Jun 27 '12 at 19:33
Thanks for the advice, I thus posted that question on MO too... –  AlexPof Jun 27 '12 at 19:58
Can you please post a link to the MO question? –  Rasmus Jun 28 '12 at 8:44
Ok, I edited the question with a link... –  AlexPof Jun 28 '12 at 9:12