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(This is a cross-post from MathSE, as someone remarked that the question would be more appropriate on MO)

Fiore and Leinster have proved that if $\mathcal{A}$ is a free monoidal category freely generated by one object $A$ such that there exists and 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 $\alpha: A \otimes A \to A$ (is not necessarily an isomorphism)isomorphism, and that there exist 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 $F \subset Aut(X)$. Loosely speaking, my approach is that since every element of $F$ can be represented as a pair $(R,S)$ of forests, we can always represent $R$ by replacing the a suitable composition of $\beta$ maps, then $\alpha^{-1}$ in Fiore's text S$by a composition of$\beta$but this is as far as I went...\alpha$ maps, the identity $\alpha \circ \beta = id$ ensuring that every facing caret gets cancelled to form a reduced forest diagram, i.e a unique element of $F$.

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Thompson's group F and monoidal categories

(This is a cross-post from MathSE, as someone remarked that the question would be more appropriate on MO)

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...