If $T$ and $S$ are two monads on the category of sets $Set$ with isomorphic algebra categories, i.e. Eilenberg-Moore categories (see https://ncatlab.org/nlab/show/Eilenberg-Moore+category), in what way can we then relate the monads themselves? For example, are they isomorphic as endofunctors? Do they have similar properties?
I ask this question because i am interested in cartesian monads and I would like to know if this property transfers. There are a lot of sources on monads, but I havent seen any that talk about this, so if anyone has a good reference, that is also much appreciated!