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That follows from the general construction that associates a monad to a Lawvere theory. If memory serves well you can find the details of this construction on this paper by Hyland and Power. Hope th …

Well the two monads are quite different: in May definition you deal with an actual monad in $\mathbf {Cat}$ (i.e. a strict-$2$-category) while in the second case you work with monads in the bicategory … Nonetheless the two monads are actually linked together: May's monad is the monad functor part used to build the algebras of Leinster's $T$-operad. Here follows the details of the construction. …

In particular in Eduardo Pareja Tobes answer are linked some papers that shows a correspondence between respectively finitary monads and Lawvere's algebraic theories and monads with arieties and theories … So it seems that there almost every time a correspondence between theories (here by theories I mean the categorical presentation of a theory given by its syntactic category) and monads of some sorts. …

Leinster proved that there are different operads from which arise the same monad, in this way he proved that operads cannot be identified with monads. …

I am not aware of Kock's works. Nevertheless Kelly provides the definition of its tensor product in the next page: it defines its tensor product $\mathcal A \otimes_{\mathcal F} \mathcal B$ as the $\ …

I've begun to interest in algebraic theories and their categorical models: in particular monads, generalized multicategories and operads, lawvere theories and their generalization. …