The following correspondence between algebraic theories and monads on $\mathbf{Set}$ is well-known (see, for example, *Algebraic Theories: A Categorical Introduction to General Algebra*).

The category of (finitary) $S$-sorted algebraic theories is equivalent to the category of (finitary) monads on $\mathbf{Set}/S$. The category of models for a fixed algebraic theory $T$ is equivalent to the category of algebras for the corresponding monad.

I would like to know if there is a known analogous result in the setting of essentially algebraic (i.e. finite limit) theories. That is, some statement like the following.

The category of (finitary) $S$-sorted essentially algebraic theories is equivalent to the category of [some class of] monads on [some category]. The category of models for a fixed algebraic theory $T$ is equivalent to the category of algebras for the corresponding monad.

There are many characterisations of categories of models of essentially algebraic theories (i.e. locally presentable categories), but I have not been able to find one in terms of a category of algebras for some monad. Given the many generalisation of Linton's result, for example in *Notions of Lawvere theory* and *Monads with arities and their associated theories*, it would seem like this result should be a straightforward application of an existing theorem, but, as far as I can tell, the case of essentially algebraic theories is never explicitly treated. (I originally read Theorem 6.7 of *Notions of Lawvere Theory* as stating that one-sorted essentially algebraic theories *also* correspond to finitary monads on $\mathbf{Set}$, but this seems unlikely to be correct.)

If the $S$-sorted case is unknown, I am also interested in the correspondence specifically in the one-sorted setting.

abeliangroups" in my previous comment. $\endgroup$ – Robert Furber Feb 22 at 20:50