Adjoint functor theorems for lax-idempotent pseudomonads

Abstract

For each pair of lax-idempotent pseudomonads $R$ and $I$, for which $I$ is locally fully faithful and $R$ distributes over $I$, we establish an adjoint functor theorem, relating $R$-cocontinuity to adjointness relative to $I$. This provides a new perspective on the nature of adjoint functor theorems, which may be seen as methods to decompose adjointness into cocontinuity and relative adjointness. As special cases, we recover variants of the adjoint functor theorem of Freyd, the multiadjoint functor theorem of Diers, and the pluriadjoint functor theorem of Solian--Viswanathan, as well as the adjoint functor theorems for locally presentable categories. More generally, we recover enriched $Φ$-adjoint functor theorems for weakly sound classes of weight $Φ$.