I am trying to understand the equivalence between the 2 category of braided G crossed categories and the 2 category of braided categories containing Rep(G) as a symmetric category. The references in this important paper:

http://arxiv.org/pdf/0906.0620.pdf

brought me to this paper by Alexander Kirillov Jr:

http://arxiv.org/pdf/math/0110221.pdf

which also has references to some previous papers by the same author.

I am not familiar with graphical calculus since I don't work in this field. I would really appreaciate if one can write down for me the formulae for the morphism T_X from Formula 4.5 on page 14 of this paper.

I also have the following question:

What is the G crossed braiding $X\otimes_A Y \rightarrow \;^gY\otimes X$ if $X \in Rep_g(A)$ in the settings of this paper?

Thank you in advance for your help!