Imagine a monoidal functor between ribbon categories (i.e. monoidal, with a braiding, a twist and compatible left and right duals). An important example would be the restriction functor from the representation category of one quasitriangular quantum group $G$ to a quantum subgroup, $H$.


Are there functors that don't preserve the braiding (this would be a very strong assumption), but preserve the duals?


I would call such a functor a "rigid functor", but apparently this term isn't being used much.
Such functors would preserve all unbraided planar diagrams, but not necessarily the braided ones.


Are there easy-to-check criteria for this in the case of quantum groups? Has it been studied (e.g. with a list of known examples)?



When does a monoidal functor between ribbon categories preserve cups and caps, but not necessarily braidings?
For Rep($U\_q(g)$), the monoidal and pivotal structures can be defined with coefficients depending only on $q$, while the braiding requires fractional powers of $q$, say $q^{1/m}$, where $m$ depends on the Lie algebra $g$. So one source of examples would be to replace $q^{1/m}$ with $\xi q^{1/m}$, where $\xi^m = 1$. This changes the braiding but doesn't affect the monoidal structure or the pivotal (duality) structure. In other words, the "identity" functor is monoidal and preserves duals, but does not preserve the braiding.


