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.

PS On a second thought, the matter of monadic vs. equational presentation just complicates things here. If we begin by presenting this algebraic structure, carried by all free T-algebras, as another monad, say (S,\eta,\mu), the proof boils down to two diagrams. The assumption that each free T-algebra is an S-algebra means that there is a natural transformation s:ST->T: its components are the given S-algebras on TX; the assumption that each Tf preserves the structure is just the naturality of s. On the other hand, the assumption that \mu of T preserves it means that \mu s = s S\mu. Using this and the naturalities, one directly checks that if a:TX->X is a T-algebra, then SX --S\eta--> STX --s--> TX --a--> X must be an S-algebra --- clearly preserved by T-morphisms. All the best, -- Dusko