Let $f: X \to Y$ be a morphism of ringed spaces. Fix a $\mathcal{O}_Y$-module $\mathcal{G}$ and a $\mathcal{O}_X$-module $\mathcal{F}$.
It is well known that the fuctors $f^*, f_*$ are adjunct via the adjunction formula
$$Hom_{\mathcal{O}_X}(f^*\mathcal{G},\mathcal{F})= Hom_{\mathcal{O}_Y}(\mathcal{G}, f_*\mathcal{F})$$
This gives arise for counit and unit maps
$$c: f^* f_*\mathcal{F} \to \mathcal{F}$$
$$u: \mathcal{G} \to f_*f^*\mathcal{G}$$
of $\mathcal{O}_X$-modules (resp $\mathcal{O}_Y$-modules).
My question is if there are some criterions for $X$ when $c$ is an isomorphsim of $\mathcal{O}_X$-modules?
(resp " for $Y$ " s.t. $u$ " )
I have some properies like projectiveness, flatness or similar in mind... if not too effortful I would be glad to references or proof sketches (or maybe short ideas) why it works
sure it suffice to prove the isomorphism on level of stalks