TITLE: Adjoint functor theorems on 2-categories
QUESTION [5 upvotes]: Adjoint functor theorems are theorems stating that under certain conditions a functor that preserves limits is a right adjoint, and a functor that preserves colimits is a left adjoint. (from the nLab.)

Now, the condition that a functor is a left/right adjoint can be stated without any further assumption on a generic 2-category. The condition that a functor preserves co/limits can be stated provided the 2-category we live in has a Yoneda structure. Of course, the co/limit must be weighted.

Is there an adjoint functor theorem that can be stated in a 2-category with a Yoneda structure?

In particular,

is there something resembling a "solution set condition" that can be tested formally?

Is such a question meaningful? What can be said, and what is an interesting open problem?

REPLY [2 votes]: There is actually such a theorem in the original Street-Walters paper on Yoneda structures (p372): any colimit-preserving functor from a total object to an admissible object has a right adjoint.  (This is a "special" adjoint functor theorem, i.e. it imposes "reasonability" conditions on the categories rather than on the functor.)