A simple characterization of theories of varieties

We answer the question of which theories in the sense of Gabriel and Ulmer give rise to (finitary, multi-sorted) varieties and exhibit the 2-category of varieties as a (non-full) bireflective sub-2-category of the 2-category of locally finitely presentable categories. A central role is played by tho se objects, which we call effective projectives, whose hom functors preserv e coequalizers of reflexive graphs, (C) 2000 Academic Press.