In learning about adjoint functors $C \underset{R}{\overset{L}{\rightleftarrows}} D$, we learn about an isomorphism of bifunctors $$\text{Hom}_D(L(\cdotp), \cdotp) \simeq \text{Hom}_C(\cdotp, R(\cdotp)).$$
Suppose we call this isomorphism $\theta$, and suppose we want to derive from $\theta$ the unit and counit. The way my book does it is, given $X \in C$, take the map that takes $X$ to $\theta_{X,LX}(\text{Id}_{LX}) \in \text{Hom}_{C}(X, RLX)$. My book now claims that this morphism in $C$ is natural in $X$. I suppose they mean that for $f \in \text{Hom}_C(X,Y)$, the following diagram commutes:
$$\require{AMScd} \begin{CD} X @>\theta_{X,LX}(\text{Id}_{LX})>> RLX \\ @VfVV @VVRL(f)V\\ Y @>\theta_{Y,LY}(\text{Id}_{LY})>> RLY \end{CD}$$
How can I see that this commutes? Presumably it comes from the naturality of $\theta$ somehow...
Roughly Both compositions from $X$ to $RLY$ in the diagram agree with $\Theta_{X,LY}(L(f))$ as follows from the naturality of $\Theta$ in both variables.
Some detail For example, the naturality of $\Theta_{X,-} : \text{Hom}_{\mathcal D}(LX,-)\cong\text{Hom}_{\mathcal C}(X,R(-))$ gives $R(Lf)\circ\Theta_{X,LX}(\text{id}_{LX})=\Theta_{X,LY}(Lf\circ\text{id}_{LX})=\Theta_{X,LY}(Lf)$; for this, note that the LHS is the image of $\Theta_{X,LX}(\text{id}_{LX})\in\text{Hom}_{\mathcal C}(X,RLX)$ under the map $R(Lf)\circ -: \text{Hom}_{\mathcal C}(X,RLX)\to\text{Hom}_{\mathcal C}(X,RLY)$ induced by $Lf: LX\to LY$ through the functor $\text{Hom}_{\mathcal C}(X,R(-))$.