Let $\mathcal{M}$ be a braided monoidal category (BMC) with braiding $\gamma$. In the definition of a BMC $\gamma$ is required to satisfy the two hexagon identities. However since "braided" appears in the name, I would have expected that $\gamma$ instead was required to satisfy the braid equation $$ (\gamma \otimes id) \circ (id \otimes \gamma) \circ (\gamma \otimes id) = (id \otimes \gamma) \circ (\gamma \otimes id) \circ (id \otimes \gamma). $$ Is it possible that this is a consequence of the hexagon equations?
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Indeed, the axioms of a braided monoidal category are enough to derive the Yang-Baxter equation. See Braided monoidal categories by Joyal and Street (diagram B7), or 1Lab for a formalised proof.
