Monad morphism ($\alpha :S\Rightarrow T$)

A morphism of monads on one category is a natural transformation $\alpha :S\Rightarrow T$ with $\alpha {\eta }^S={\eta }^T$ and $\alpha {\mu }^S={\mu }^T(T\alpha )(\alpha S)$. These equations preserve the unit and multiplication. A terminal endofunctor in a full composition-closed subcategory containing the identity receives a unique such morphism from every monad whose functor belongs to that subcategory.

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.