According to categories:
Date: Wed, 01 Feb 1995 11:51:57 +0000 From: Andrea Schalk <Andrea.Schalk@cl.cam.ac.uk>
I'm looking for some pointer to literature where I can find the following (or a similar) Theorem:
Let $(T,\eta,\mu)$ be a monad. Assume we have an algebraic structure such that every free algebra for that monad carries one of those algebraic structures and such that all morphisms of the form $Tf$ and all $\mu_C$ preserve it. Then all Eilenberg-Moore algebras carry such a structure and all morphisms between them preserve it.
Manes' book "Algebraic Theories" contains many propositions and exercises of this kind, relating monadic and equational presentations of algebraic theories. I think something of this kind should be there. Regards, -- Dusko Pavlovic