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 algebraic generator, i.e., a regular generator which is exactly projective and finitely generated. This improves somewhat a recent restatement, due to Pedicchio and Wood, of the classical characterization theorem of Lawvere. A bijective correspondence between algebraic theories and algebraic generators is established.

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