Adjoint Functors induce Natural Transformations Let $L:\mathcal{C} \to  \mathcal{D}$ and $R:\mathcal{D} \to  \mathcal{C}$ to adjoint functors, therefore for each $X \in \mathcal{C}, Y \in \mathcal{D}$ we have natural bijections
$$Hom_{\mathcal{D}}(L(X), Y) \cong Hom_{\mathcal{C}}(X, R(Y)) $$
My question is how to see that these induce natural transformation $\eta: id_{\mathcal{C}} \to RL$ (resp. $\epsilon: id_{\mathcal{D}} \to LR$)?
Here I don't see how to use the fact that $R$ and $L$ are adjoint to see that for arbitrary morphism $f: X \to X'$ in $\mathcal{C}$ following diagram commutates (therefore "natural transformation"):
$$
\require{AMScd}
\begin{CD}
X  @>{\eta_X}  >> RL(X)   \\
@VVfV  @VVRL(f)V   \\
X' @>{\eta_{X'}}>> RL(X') 
\end{CD}
$$

Hint: In particular, you have $$\operatorname{Hom}_{\mathcal D}(LX,LX)\cong\operatorname{Hom}_{\mathcal C}(X,RLX).$$
Can you think of a particular morphism on the left side you could send to the right side via this isomorphism?
