Equational presentations of functors and monads

In categorical universal algebra, it is well known that a finitary equational presentation of algebras in a finitary variety K amounts to the existence of a coequaliser of a pair of morphisms between finitely generated free algebras. The essence of a universal-algebraic flavour of a finitary functor L : K −→ K is that L is determined by its behaviour on finitely generated free algebras. In fact, such functors also admit an equational presentation, but this time the coequaliser is more complex, though it again involves functors freely generated from Set-functors. Recently, such functors L have appeared naturally in the study of modal algebras for coalgebraic modal logic, see, for example, Bonsangue and Kurz (2006) or Kurz and Rosický (2006). In fact, for the case of one-sorted varieties K , functors L that are determined by their values on finitely generated free algebras are exactly the functors preserving a class of colimits and referred to as sifted or, equivalently, they are exactly the class of functors admitting an equational presentation (Kurz and Rosický 2006). In the current paper we study presentations of functors/monads on K that are determined by finitely generated free algebras, but we want the requirements on K to be as relaxed as possible in view of Kurz et al. (2010). Hence we again study functors/monads L : K −→ K , but K is now only required to be a full subcategory of a variety, though K must still contain free objects on finitely many generators. Thus, the initial setting is now given by a finitary adjunction F U : K −→ X that is of descent type, that is, such that K embeds fully into the Eilenberg–Moore category