= Restriction of monad algebras
{title2=$(A,a:TA\to A)\longmapsto(A,a\alpha_A:SA\to A)$}
A <monad morphism> $\alpha:S\Rightarrow T$ induces a <functor> $\mathcal C^T\to\mathcal C^S$ between <Eilenberg-Moore categories>, identical on underlying objects and arrows. Its unit equation gives the restricted algebra's unit law. Naturality and the multiplication equation give $(a\alpha_A)S(a\alpha_A)=aTa\,\alpha_{TA}S\alpha_A=a\mu_A^T T\alpha_A\alpha_{SA}=a\alpha_A\mu_A^S$. Naturality also preserves algebra <morphisms>.
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