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