Unless I'm mistaken the "Free completion under finite limits monad" $C \mapsto C^{lex}$ and the "free co-completion monad" $C \mapsto \widehat{C}$ (the categories of small presheaves) satisfies a distributivity law, and the algebra for the composed monad $C \mapsto \widehat{C^{lex}}$ are the infinitary pretopos.

Has this been written out somewhere ? I have a vague memories of seeing something like that some time ago (or maybe it was for completion under co-products and extensive categories), but I have not been able to find it anywhere...

Is there a similar statement for finitary pretoposes ?

I assume $C \mapsto \widehat{C}$ has to be replaced by completion under finite co-products and some co-equalizer, but that seems to be a little bit more tricky. I guess in technical terms what I want to know is: "is it possible to write the free pre-topos monad as the composite of a co-KZ-monad and a KZ-monad satisfying a distributivity law ?"

exists(and is unique because it's between a lax-idempotent and a colax-idempotent 2-monad). The last section of A classification of accessible categories treats a similar family of distributive laws, and at least says something about the algebras for the composite 2-monads. $\endgroup$