Consider a unitary modular TQFT, defined by the F and R moves. More specifically, a braided tensor category relevant for anyon models in 2D topologically ordered phases of matter. I am interested in the value of the F-move, $[F^{abc}_1]_{\bar{a}\bar{c}}$, diagrammatically defined by where $1$ is the vacuum and $\bar{a}$ is the inverse of $a$, i.e. $1 \in \{a\times \bar{a}\}$

I would naively expect this to be the identity. However, in https://thesis.library.caltech.edu/2447/2/thesis.pdf the author states that this need not be the case.

I think this is related to bending, and/or the $\mathbb{Z}_2$, or possibly the $\mathbb{Z}_3$, Frobenius Schur indicator. Any clarification (or references) on the value of this F-move would be much appreciated!