Assume we have a braided pivotal monoidal category. This means we assume the braiding $c$ to be a natural isomorphism. But looking at the corresponding string diagram, it seems to me as if we could obtain $c^{-1}$ as the composition of the following morphisms (treating the monoidal structure as strict): $$ A \otimes B \xrightarrow{\eta_{B^{*}} \otimes A \otimes B} B^{**} \otimes B^{*} \otimes A \otimes B \xrightarrow{B^{**} \otimes c_{A, B^{*}} \otimes B} B^{**} \otimes A \otimes B^{*} \otimes B \xrightarrow{B^{**} \otimes A \otimes \varepsilon_{B}} B^{**} \otimes A = B \otimes A $$

As a string diagram, this construction would look like this:

This begs the questions:

Is it enough to assume the braiding as a morphism instead of assuming it to be an iso?

When working 2-categorically, we now have a non-trivial 2-cell at $c^{-1}$, should this be filled to make the notion well-behaved?

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