Let $f:X \to Y$ be an affine morphism; is it true that the counit map


\begin{equation\*} f^\* f\_\* \mathcal{F} \to \mathcal{F}
\end{equation\*}
is surjective for every (coherent) sheaf $\mathcal{F}$?



Surjectivity counit of pushforward-pullback adjunction of an affine morphism
Yes. Since $f$ is affine, it is enough to check surjectivity after applying $f\_\*$. But the composition
$$ f\_\* F \to f\_\* f^\* f\_\* F \to f\_\* F $$
of $f\_\*({\rm counit})$ with the unit of $f\_\* F$ is the identity (triangle identities). 


