The tangle hypothesis, when specialized to ordinary framed tangles, says that the framed tangles form the free braided category with all duals (i.e. considered as a 3-category, all the 1- and 2-morphisms have adjoints) on one object. Another characterization of framed tangles is by ribbon categories. This seems to suggest that a ribbon category is the same as a braided category with all duals. Is this true?
If this is so, then since having adjoints is a property, the twist in the definition of ribbon categories must be unique, because there can't be more data here. Either this is secretly the case (which is pretty surprising), or the two concepts are different. It should furthermore imply that the ribbon element in ribbon hopf algebras act in the same way on all the finite-dimensional representations, which is most probably not true... Is there a proof or reference confirming either of these? If the latter, is there an equivalent definition of a braided category with all duals, removing all the redundant components and translated to purely 1-categorical terms? This should be an alternative way to create Reshetikhin–Turaev knot invariants.