Under which assumptions counit of the adjunction $f^* f_* \to 1$ is epimorphic? Let $f: X \to Y$ be a morphism of schemes. It produces a pair of adjoint functors $f^*$ and $f_*$ on the category of quasi-coherent sheaves i.e. there is a natural isomorpism
$$
\operatorname{Hom}(f^*S_1,S_2) \cong \operatorname{Hom}(S_1,f_*S_2),
$$
for any quasi-coherent $S_1$ and $S_2$.
Under which assumptions the counit of the adjunction
$$
f^* f_* S \to S
$$
is an epimorphism?
I need this property for coverings of smooth projective curves only, but general picture is also interesting.

$f^* f_* M \to M$ is an epimorphism iff $\hom(M,N) \to \hom(f^* f_* M,N)$ is injective for all $N$ iff $\hom(M,N) \to \hom(f_* M,f_* N)$ is injective for all $N$. Hence, $f^* f_* M \to M$ is an epimorphim for all $M$ iff $f_*$ is faithful. This argument works for any adjunction.
Now, for which morphisms $f$ is $f_*$ faithful? It is the case when $f$ is an immersion (in which case actually $f_*$ is fully faithful, which means that $f^* f_* M \to M$ is an isomorphism). It is also satisfied when $f$ is affine (then $f_*$ identifies with a forgetful functor), in particular for finite morphisms, in particular for finite coverings.
