Supposedly the category $Bord^{\operatorname{or}}_{1,2,3}$ carries with it the structure of a Hopf algebra. If that's the case I would like to understand what the antipode is.
To that end, I've been reading this paper. On page 19 diagram 36 seems to be similar to that of the antipode relation of a Hopf algebra however it is called the modularity condition in the paper. The antipode relation is shown as the very first diagram at this wiki page.
So, my question is, how are the modularity relation and the antipode related?