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!