For monads on categories, the kleisli category can be described via kleisli composition, as free algebras, or via a perhaps less well-known generators and relations presentation. In this setting, these are all isomorphic categories. Moreover, the canonical comparison functor from the kleisli category to any other adjunction splitting the same monad is fully faithful. Thus such an adjunction has the universal property in the up to equivalence sense if and only if it's left adjoint is essentially surjective on objects.
All three of these constructions can be categorified to the two-dimensional setting but only the last one produces a 2-category with appropriate the Gray-categorical universal property. Indeed, the kleisli composition construction only categorifies to give a bicategory while with the 2-category of free pseudoalgebras one only gets a pseudofunctor FreePsAlg(A, T) → B from a pseudoalgebra for Gray(T, B): Gray(A, B) →Gray(A, B).
The main theorem of the talk will be that the canonical comparison 2-functor from the Gray-categorical kleisli object to any other pseudoadjunction splitting the same pseudomonad is bi-fully-faithful (ie is given by equivalences between hom-categories). With this we will see that all three constructions are biequivalent, greatly simplifying the presentation. We then deduce that a pseudoadjunction is kleisli in the up to biequivalence sense if and only if it's left pseudoadjoint is biessentially surjective on objects. Armed with a property that is closed under composition, we conjecture a description of a tricategorical 'free cocompletion under kleisli objects for pseudomonads', as well as a seemingly simpler 'free completion under Elienberg Moore objects for pseudomonads'.