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(When) does a morphism of monad induce adjoint functors between categories of algebras?

For monads $S$ and $T$ on a fixed Abelian category $C$, a morphism of monads $\sigma: S\rightarrow T$ induces a functor between Eilenberg-Moore categories $\sigma^*:C^T\rightarrow C^S$. … Monads $S$ and $T$ are induced by an adjoint pair, there is a canonical $free \dashv forgetful$ adjunction for EM categories, so I would like to have an adjoint for $\sigma^*$ too. …
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