Equalizer submonad of a monad (${\beta }_A:RA⟶TA$)

In a category with equalizers, the equalizer of ${\eta }_{TA},T{\eta }_A$ defines a functor $R$ by ${\beta }_{B}Rf=Tf{\beta }_A$. It has unit $\alpha$ and multiplication $\theta$ characterized byCancelling the monomorphisms ${\beta }_A$ reduces its unit and associativity laws to those of $T$, so $\beta :R\Rightarrow T$ is a monad morphism. The split equalizers associated with a monad show that ${\alpha }_{TA}$ is invertible. If every $T{\beta }_A$ is monic, then every $T{\alpha }_A$ is invertible, the naturality square of $\beta$ at ${\beta }_A$ is a pullback in a category, and $R$ is an idempotent monad. This monicity hypothesis is essential to this last argument; arbitrary functors need not preserve monomorphisms.