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> 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](https://ncatlab.org/nlab/show/adjoint+functor+theorem)*.)
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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.



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



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> is there something resembling a "solution set condition" that can be tested *formally*?
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Is such a question meaningful? What can be said, and what is an interesting open problem?



Adjoint functor theorems on 2-categories
There is actually such a theorem in the original Street-Walters paper on Yoneda structures (p372): any colimit-preserving functor from a [total object](https://ncatlab.org/nlab/show/total+category) 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.)


