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 $\varepsilon$.
 A: There is a useful trick for dealing with this.
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.
