so consider "the free symmetric monoidal category on one invertible object with trivial self-braiding" ... martin asks about whether the inverse object here has the same property ...???

??something about... ???taking inverse as contravariant symmetric monoidal equivalence on the invertible objects?? ...??or something?? ... ??something about mates??

hmmm... ???mate of identity morphism as identity morphism ... ??only if... you're careful to "use the same inverse" on both domain and co-domain ?? ... ??or something ???...

??what about something about ... "adjoint equivalence" here, or something... ??was that supposed to be different somehow from an ordinary equivalence??? ... sounds weird... ??maybe that issue is a level-slip away???...

??or maybe it's _not_ a level-slip away???

??inverse objects vs adjoint-inverse objects??