Assume $X$ and $Y$ are noetherian schemes over $\mathbb{C}$ and there is a proper and faithfully flat morphism $f: X\rightarrow Y$.


Assume the canonical morphism $F\xrightarrow{\sim} f\_{\*}f^{\*}F$ is an isomorphism for all $F\in Coh(X)$.


Can we say anything about the canonical morphism $f^{\*}f\_{\*}f^{\*}F\rightarrow f^{\*}F$ $(\*)$? Is this also an isomorphism?


We have an isomorphism $f^{\*}F\xrightarrow{\sim} f^{\*}f\_{\*}f^{\*}F$ if we just pullback the canonical isomorphism. Does this isomorphism have anything to do with $(\*)$?


Are there any conditions under which we can deduce that $F\xrightarrow{\sim} f\_{\*}f^{\*}F => f^{\*}f\_{\*}f^{\*}F \xrightarrow{\sim} f^{\*}F$?


Is there maybe a categorical argument here, since we speak about adjoint functors?



About the canonical morphism from $f^{*}f_{*}f^{*}F$ to $f^{*}F$
In short: always.


Indeed, given a functor $F : \mathcal{C} \to \mathcal{D}$ left adjoint to $G : \mathcal{D} \to \mathcal{C}$, the triangle identities say that the composites of the canonical morphisms
$$F X \to F G F X \to F X$$
$$G Y \to G F G Y \to G Y$$
are identities for all $X$ in $\mathcal{C}$ and all $Y$ in $\mathcal{D}$. In particular, if $X \to G F X$ is an isomorphism, then so is $F G F X \to F X$.


