# Profunctors as a Kleisli bicategory

There is some discussion on the nLab on seeing the free cocompletion $$\mathbf{Psh}(\mathbf{A}) = [\mathbf{A}^{op}, \mathbf{Sets}]$$, as a pseudomonad. The Yoneda embedding $$よ \colon \mathbf{A} \to \mathbf{Psh}(\mathbf{A})$$ gives the unit and that the multiplication would be the unique functor $$\mu \colon \mathbf{Psh}(\mathbf{Psh}(\mathbf{A})) \to \mathbf{Psh}(\mathbf{A})$$ such that $$\mu \circ よ = \mathrm{id}$$. However, this creates some size issues.

On the other hand, composition of profunctors $$p \colon \mathbf{B} \to \mathbf{Psh}(\mathbf{A})$$ and $$q \colon \mathbf{C} \to \mathbf{Psh}(\mathbf{B})$$ can be defined (following Remark 5.6 in This is the (co)end...) as $$\mathsf{Lan}_よ(p) \circ q$$, which looks like composition in a Kleisli category for the (pseudo)monad.

Can we get the bicategory of profunctors as a Kleisli construction over the free-completion pseudomonad? Can we deal somehow with the size issues? Is there any reference where this construction is discussed?

• out of curiosity, what goes wrong if you fix a cardinal kappa and try to execute this plan for kappa-accessible categories, kappa-accessible presheaves, etc.? Jun 26 '19 at 23:10

This beautiful story take place in $$\mathsf{Cat}$$, the bicategory of locally small categories. There, the construction of small presheaves induces a pseudomonad whose pseudoalgebrabas are cocomplete categories and whose Kleisli category is $$\mathsf{Prof}$$. A part of this story is folklore and is partially discussed in Lex Colimits, by Garner and Lack, Prop 2.2.