Let $f:R\to S$ a ring homomorphism and $f^* \dashv f_*$ the restriction-coinduction adjunction. I know that $f_*$ is faithful iff the counit $\epsilon_V$ of this adjunction is an epimorphism for all $V$. Is there an easier condition on $f$ for this to be true?
For example, I know that the restriction functor $f^*$ is full iff $f$ is an epimorphism. I find it tempting to believe that $f_*$ is faithful iff $f$ is mono but I have absolutely no evidence for that.
The counit $\epsilon_V$ is the $R$-module homomorphism $$\text{Hom}(f,V):\text{Hom}_R(S,V)\to\text{Hom}_R(R,V)=V.$$ For this to be epi for all $V$ we need $f$ split mono as an $R$-module homomorphism (take $V=R$ and $\text{id}_R\in\text{Hom}_R(R,R)$). This is also a sufficient condition.