Lawvere theories enriched over a general base

We generalise the correspondence between Lawvere theories and finitary monads on Set in two ways. First, we allow our theories to be enriched in a category VV that is locally finitely presentable as a symmetric monoidal closed category: symmetry is convenient but not necessary. And second, we allow the arities of our theories to be finitely presentable objects of a locally finitely presentable VV-category AA. We call the resulting notion that of a Lawvere AA-theory. We extend the correspondence for ordinary Lawvere theories to one between Lawvere AA-theories and finitary VV-monads on AA. We illustrate this with examples leading up to that of the Lawvere Cat-theory for cartesian closed categories, i.e., the Set-enriched theory on the category Cat for which the models are all small cartesian closed categories. We also briefly investigate change-of-base.