I have to do some messy calculations with weak 2-functors between bicategories, and I know the most efficient way to do it would be via some sort of string diagram methods. Also, it means that I can put pretty calculations in my paper, unlike some of this paper's 'ancestors' (written by others) where they simply can't put in all the diagrams. To me this seems to be a blow to the reader, because not only does the reader have to figure out what on earth is going on, they are stuck with inelegant methods for doing so.
I know exactly what I should be doing if I was working in a single bicategory (just a simple extension of string diagrams for monoidal categories) but I don't know if anyone has written out a version with weak 2-functors. I'm sure someone has. Any pointers?