Suppose we have adjoint functors $C \underset{R}{\overset{L}{\rightleftarrows}} D$ with adjunction $$\theta: \text{Hom}_D(L(\cdotp), \cdotp) \xrightarrow{\sim} \text{Hom}_C(\cdotp, R(\cdotp)).$$
Let $\epsilon: \text{Id}_C \to R \circ L$ be the unit and $\eta: L \circ R \to \text{Id}_D$ be the counit. My book claims $$(\eta \circ L) \circ (L \circ \epsilon): L \to L \circ R\circ L \to L \,\,\text{ is Id}_L. \tag{1}$$ $$(R \circ \eta) \circ (\epsilon \circ R): R \to R \circ L\circ R \to R \,\,\text{ is Id}_R \tag{2}.$$
In trying to prove (1), for example, I want to prove that for all $X \in C$, we have $L(\epsilon_X) = {\eta_{LX}}^{-1}.$ I think this has to come from the naturality of $\theta$. This is my first time using naturality to show that two maps are inverses, so I'm stumbling a bit. Hunting around for the answer, I sketched the following incomplete diagram:
I want to show that if we start with $\text{Id}_{LX}$ in the upper-left hand corner, we get $\text{Id}_{LX}$ in the bottom-right corner. I note that if we go counter-clockwise, we end up with $\eta_{LX}$. But I'm sort of stuck. Can anyone help?
Once you have extracted the counit $\eta: LR\Rightarrow\text{Id}$ from the adjunction isomorphism $\Theta^{-1}$, the latter can be recovered as follows:
Given $f: X\to RY$, its adjoint $\Theta^{-1}_{X,Y}(f): LX\to Y$ is given by the composit $$LX\stackrel{Lf}{\longrightarrow} LRY\xrightarrow{\eta_{Y}=\Theta_{RY,Y}^{-1}(\text{id}_{RY})}Y;$$ this follows from an application of naturality of $\Theta^{-1}$ in first variable and is in fact an instance of the Yoneda lemma applied to the natural transformation $\Theta^{-1}_{-,Y}: \text{Hom}_{\mathcal C}(-,RY)\to\text{Hom}_{\mathcal D}(L(-),Y)$.
In particular, taking $Y=LX$, the composition $$\text{Hom}_{\mathcal C}(X,RLX)\xrightarrow{L}\text{Hom}_{\mathcal D}(LX,LRLX)\xrightarrow{\eta_{LX}\circ-}\text{Hom}_{\mathcal D}(LX,LX)$$ in your diagram equals $\Theta^{-1}_{X,LX}$, as required.
Recovering $\Theta$ from $\varepsilon:\text{Id}\Rightarrow RL$ works similarly.