We give a new account of the correspondence, first established by Nishizawa--Power, between finitary monads and Lawvere theories over an arbitrary locally finitely presentable base. Our account explains this correspondence in terms of enriched category theory: the passage from a finitary monad to the corresponding Lawvere theory is exhibited as an instance of free completion of an enriched category under a class of absolute colimits. This extends work of the first author, who established the result in the special case of finitary monads and Lawvere theories over the category of sets; a novel aspect of the generalisation is its use of enrichment over a bicategory, rather than a monoidal category, in order to capture the monad--theory correspondence over all locally finitely presentable bases simultaneously.

• 18C10 - Theories (e.g., algebraic theories), structure, and semantics
• 18C20 - Eilenberg-Moore and Kleisli constructions for monads
• 18C35 - Accessible and locally presentable categories
• 18D20 - Enriched categories (over closed or monoidal categories)

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