Why does the definition of a braided monoidal category not mention the braid equation?

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?