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Looking for a reference for the following easy-to-prove fact:

Say $T$ and $S$ are monads on $\text{Set}$ admitting a monoid homomorphism $\phi : S \to T$ (i.e., a morphism in $\text{Mon}([\text{Set},\text{Set}])$. Then any $T$-algebra is also an $S$-algebra.

I imagine this might be stated in the generality of monoids/module objects in a monoidal category $\mathcal{C}$; the statement above may also be missing some condition on $\phi$. I have checked pretty much every elementary textbook on category theory I can think of, and I find it nowhere. Thanks in advance!

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    $\begingroup$ That phrasing is a little ambiguous; it seems clearer to say that pullback along $\phi$ defines a functor from $T$-algebras to $S$-algebras, etc. Even in familiar examples there may be more than one candidate for $\phi$, e.g. there are two homomorphisms from the monoid monad to the ring monad, corresponding to addition and multiplication. $\endgroup$
    – Qiaochu Yuan
    Commented Dec 3 at 4:45
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    $\begingroup$ This was asked last year under different terminology: The change-of-monoid adjunction between categories of modules induced by a morphism of monoids. (Though if you just want the monad case, that’s a bit more specific and might be out there somewhere else.) $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Dec 3 at 7:57
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    $\begingroup$ It is certainly mentioned matter-of-factly as completely well known in "Coequalizers in categories of algebras" of Linton (link.springer.com/chapter/10.1007/BFb0083082). One may argue that this is already in "Functorial semantics of algebraic theories" of Lawvere (pnas.org/doi/abs/10.1073/pnas.50.5.869), though of course there it is mentioned (as obvious) for equational algebraic theories only. $\endgroup$
    – Vladimir Dotsenko
    Commented Dec 3 at 9:24

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I believe the original reference for this fact is Theorem 2 of Maranda's 1966 On Fundamental Constructions and Adjoint Functors, although the terminology is not modern. A more readable reference is Theorem 1 of Frei's 1969 Some Remarks on Triples, although note that Frei uses the term "triple" rather than "monad". Both references exhibit a functor $\mathbf{Alg} \colon \textbf{Mnd}(\mathbb C)^{\text{op}} \to \mathbf{CAT}/\mathbb C$ for any category $\mathbb C$.

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