Toward a characterization of algebraic exactness

It is well known that the morphisms between varieties of algebras (as objects) induced by morphisms of algebraic theories are precisely the algebraically exact functors, and they can be completely characterized as the finitary, continuous and exact functors. We prove that this characterization extends to morphisms between algebraically exact categories (forming an “equational hull” of the category of all varieties). And among categories with finite coproducts, the algebraically exact ones are proved to be precisely the precontinuous, completely exact categories.