A categorical characterization of varieties

A simple, direct proof of the following characterization of varieties of (finitary)algebras is presented: a cocomplete category is equivalent to a variety iff it has an algebraicgenerator, i.e., a regular generator which is exactly projective and finitely generated. Thisimproves somewhat a recent restatement, due to Pedicchio and Wood, of the classicalcharacterization theorem of Lawvere. A bijective correspondence between algebraic theoriesand algebraic generators is established.