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 those objects, which we call effective projectives, whose hom functors preserve coequalizers of reflexive graphs.