# Adjunction counit for sheaves is isomorphism

Let $f\colon X \to S$ be a proper morphism of varieties over $\mathbb{C}$ with $f_* \mathcal{O}_X = \mathcal{O}_S$ and $\mathcal{G}$ be a coherent sheaf on $S$. Then we have a natural morphism $\varepsilon\colon f^*f_*f^* \mathcal{G} \to f^*\mathcal{G}$ coming from adjunction (the counit transformation evaluated at $f^*\mathcal{G}$).

Why is $\varepsilon$ an isomorphism? I know that by the triangle identities, it must be a split epimorphism. Furthermore, the projection formula implies that $f^*f_*f^* \mathcal{G}$ and $f^*\mathcal{G}$ are isomorphic but I cannot find the connection of this isomorphism to $\varepsilon$.

Proposition. Given an adjunction $$L \dashv R : \mathcal{D} \to \mathcal{C}$$ if $L R \cong \mathrm{id}_{\mathcal{D}}$ (as functors) then the counit $\epsilon : L R \Rightarrow \mathrm{id}_{\mathcal{D}}$ is (also) a natural isomorphism.
Proof. Let $\delta = L \eta R$, where $\eta : \mathrm{id}_{\mathcal{C}} \Rightarrow R L$ is the unit. Then (by the triangle identities), we have a comonad: \begin{align} \epsilon L R \bullet \delta & = \mathrm{id}_{\mathcal{D}} & L R \epsilon \bullet \delta & = \mathrm{id}_{\mathcal{D}} & L R \delta \bullet \delta & = R L \delta \bullet \delta \end{align} We can transport this structure along any natural isomorphism $\theta : L R \Rightarrow \mathrm{id}_{\mathcal{D}}$, so that e.g. $$\begin{array}{rcl} L R & \overset{\theta}{\to} & \mathrm{id}_{\mathcal{D}} \\ {\scriptstyle \epsilon} \downarrow & & \downarrow {\scriptstyle \epsilon'} \\ \mathrm{id}_{\mathcal{D}} & \underset{\theta}{\to} & \mathrm{id}_{\mathcal{D}} \end{array}$$ commutes. But (using naturality) any comonad structure $(\epsilon', \delta')$ on $\mathrm{id}_{\mathcal{D}}$ must consist of natural isomorphisms, so we deduce that the original $\epsilon$ and $\delta$ are also natural isomorphisms.　◼
Now, let us reduce the claim to the above proposition. Let $\mathcal{C}$ be the full subcategory of coherent sheaves on $S$ and let $\mathcal{D}$ be the full subcategory of coherent sheaves of the form $f^* \mathscr{G}$ where $\mathscr{G}$ is a coherent sheaf on $S$. Since $f : X \to S$ is proper and both $X$ and $S$ are noetherian, both $f^*$ and $f_*$ preserve coherent sheaves. We then have a restricted adjunction $$L \dashv R : \mathcal{D} \to \mathcal{C}$$ and a natural isomorphism $L R \cong \mathrm{id}_{\mathcal{D}}$. Hence the counit is also a natural isomorphism.