I'm currently reading Richard Garner's paper Polycategories via pseudo-distributive laws, and a central construction is the lifting of the symmetric strict monoidal category 2-monad to a pseudomonad on $Prof$, the bicategory of profunctors. I'm trying to work through all the details and they are quite messy. I was wondering if there was a more general theory for this that would clean things up, like how if a monad is cartesian, it lifts to a monad on Span. 



When does a 2-functor or 2-monad of Cat lift to a psuedofunctor or pseudomonad on Prof?
Sec. 6 of [this paper](https://arxiv.org/pdf/1612.03678.pdf) should answer your question.


