Adjoint Functors induce Natural Transformations

Question

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"):

$$ \require{AMScd} \begin{CD} X @>{\eta_X} >> RL(X) \\ @VVfV @VVRL(f)V \\ X' @>{\eta_{X'}}>> RL(X') \end{CD} $$

score 2 · Accepted Answer · answered Mar 05 '18 at 20:27

2

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?

answered Mar 05 '18 at 20:27

Ennar

24,364

Ah haha, clear the $id_{LX}$ – user267839 Mar 05 '18 at 20:42
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...and the commutativity follows automatically from naturality of $Hom_{\mathcal{D}}(L(X), Y) \cong Hom_{\mathcal{C}}(X, R(Y)) $ – user267839 Mar 05 '18 at 20:45
@KarlPeter, precisely. There is no choice, really, since the identity is the only thing you know exists for sure. – Ennar Mar 05 '18 at 21:29

Adjoint Functors induce Natural Transformations

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