Gabriel-Ulmer duality and Lawvere theories enriched over a general base

Motivated by the search for a body of mathematical theory to support the semantics of computational effects, we first recall the relationship between Lawvere theories and monads on Set. We generalise that relationship from Set to an arbitrary locally presentable category such as Poset and ÉCpo or functor categories such as [Inj, Set] and [Inj,ÉCpo]. That involves allowing the arities of Lawvere theories to be extended to being size-restricted objects of the locally presentable category. We develop a body of theory at this level of generality, in particular explaining how the relationship between generalised Lawvere theories and monads extends Gabrielââ"šÂ¬Ã¢â‚¬Å“Ulmer duality.