Partially monadic functors (Q788817)

The notion of ''partially multiadjoint functor'' is an extension of the notion of ''right adjoint functor'': if \(U: {\mathcal A}\to {\mathcal B}\) is a functor and B is an object of \({\mathcal B}\), instead of looking for one universal morphism from B to U in the case of adjoint, or for a universal family of morphisms from B to U in the case of multiadjoint, the author looks for a ''universal ordered family'' of morphisms from B to U, such that any morphism from B to U factors through a greatest morphism of the family. The auther extends the notions of monads, monads generated by functors, algebras for monads, monadic functors, Beck criterium, Kleisli categories, representations by functor categories, by using partially multiadjoint functors instead of adjoint functors.