This question is related to this other question I have asked some time ago. Let $R$ and $S$ be two rings and let $\phi:R\to S$ be a ring homomorphism.

It is well-known that $\phi$ is an epimorphism in the category of rings if and only if the counit of the adjunction $(-\otimes_R{}_RS_S,-\otimes_S{}_SS_R)$ (where the structures of left or right $R$-module on $S$ are induced using $\phi$), is a natural isomorphism. I usually denote the adjunction described above by $(\phi_!,\phi^*)$, here $\phi^*:Mod(S)\to Mod(R)$ is a restriction of scalars, while $\phi_!:Mod(R)\to Mod(S)$ is extension of scalars.

Now, since $\phi_!$ is left adjoint, we can consider its restriction to the respective categories of finitely presented modules, obtaining a new functor $\Phi:=(\phi_!)_{mod(R)}:mod(R)\to mod(S)$, between the two preadditive categories $mod (R) $ and $mod (S)$. Of course, this new functor induces extension and restriction of scalars: $$\Phi_!:((mod (R))^{op},Ab)\rightleftarrows ((mod (S))^{op},Ab):\Phi^*$$ and we can ask under which conditions the counit $\Phi_!\Phi^*\to id$ is an isomorphism. This is probably the same as saying that $\Phi$ is an epimorphism in the category of preadditive small categories (with additive functors as morphisms).

My question is: Can we characterize those $\phi$ for which $\Phi_!\Phi^*\to id$ is an isomorphism? Is this equivalent to $\phi_!\phi^*\to id$ being an isomorphism, so equivalent to say that $\phi$ is a ring epimorphism, or is it a genuinely stronger condition?