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 a lot with modular tensor categories. I would really appreaciate if one can write down for me the formula (not graphical calculus) for the morphism T_X from Formula 4.5 on page 14 of this paper.

The horizontal circle probably as usually means the dimension dim X and the punctured vertical line is the identity of A.The punctured curve between the circle and the line should be the structure \mu_X of X as an A-module in C. What I don't understand is how the evaluation and coevaluation are apllied to end up again in A.

I also have the following question:

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

The braiding is given in the first paper as the unique morphism that extends the initial braiding from C. I was thinking maybe in the settings of Kirillov's paper one can write directly a formula ffor the braiding.
Thank you in advance for your help!