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I understand the statement of Yoneda Lemma and its implications; however, at a very concrete level I have never seen how naturality condition is used in examples. Even when we consider a Yoneda embedding, the naturality condition is not used to show fullness of the functor. I wonder if someone could explain why the naturality is so crucial, or perhaps provide a motivation of 'how things would be different' if it was not to hold.

To clarify, I refer to naturality in the following sense (this is taken from Tom Leinster's Basic Category Theory p95):

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This question might not sound concrete enough, compared to some popular 'confusions with Yoneda', but I think it might help for other beginners as myself. This naturality condition is somewhat implicit, and yet there are many similar instances in which naturality condition can't be ignored. For instance, when you learn about adjuctions you get to see that naturality in both arguments is constantly used to prove important results such as correspondence between co(units) and adjunctions or General Adjoint Functor Theorem. I wonder what is the importance of naturality in the case of Yoneda Lemma.

Tanizaki
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  • Can you specify what you call "the naturality condition"? – Captain Lama Jun 05 '24 at 18:48
  • @CaptainLama sure! https://en.wikipedia.org/wiki/Yoneda_lemma has a section on naturality between two functors $C\times Set^C\rightarrow Set$ – Tanizaki Jun 05 '24 at 18:59
  • To be specific, are you asking what happens in the Yoneda lemma if instead of "natural transformations" you write "transformations, not necessarily natural"? Have you worked out an example of what happens here for the case of, say, a one-object category? – Qiaochu Yuan Jun 05 '24 at 20:21
  • @Tanizaki Linking to Wikipedia is a good start but please edit your question to be more self-contained and explain what naturality condition you have in mind, given that there seems to be some confusion :) – Ben Steffan Jun 05 '24 at 20:35
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    Are you really sure you don't use naturality of Yoneda? Because it is implicitly used all the time. Every time you see "$\cong$" in category theory, you should have in the back of your mind: "the author probably means this is natural, but I need to ask myself why this is natural". It's left so implicit it gets fairly well forgotten, and stuff is assumed to work out – FShrike Jun 05 '24 at 23:27
  • So if it isn't used in your specific case, trust me it'll get used eventually. It's also just nice to know. In category theory you very rarely want unnatural isomorphisms (partly because we implicitly and sloppily assume everything is always fine... so for that to have any hope of working, you need naturality as a starting point) – FShrike Jun 05 '24 at 23:30
  • @FShrike Are you saying that the use of "$\cong$" is a kind of hand-waving which some authors believe to be justified by the magic word naturality? – Jochen Jun 06 '24 at 07:42
  • If $F$ and $G$ are functors, then $F \cong G$ means $F$ and $G$ are naturally isomorphic. If you mean something different you should specify. – Naïm Camille Favier Jun 06 '24 at 07:54
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    @Jochen not really. I’m just trying to alert the OP to the reality that people tend not to check naturality or its consequences in detail yet it is very important and frequently used! – FShrike Jun 06 '24 at 09:14
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    @BenSteffan I have edited the question, hope it's clearer now!))) – Tanizaki Jun 06 '24 at 10:32
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    Ah, so you are talking about the naturality of the Yoneda lemma itself (seen as a natural isomorphism between two evaluation functors), rather than the naturality in the hypotheses of the Yoneda lemma? Since this is just a theorem, it's not clear that you can meaningfully ask how things would be if it didn't hold, like asking how the universe would look like if $\pi$ was $4$. – Naïm Camille Favier Jun 06 '24 at 10:52

1 Answers1

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Naturality of the bijection in Yoneda Lemma is almost tautological. Take a functor $F:\mathcal{C}\to \mathsf{Set}$, an object $c$ in $\mathcal{C}$, and the map $\Phi_{c,F}: \mathsf{Set}^\mathcal{C}(\mathcal{C}(c,-),F)\to Fc$ given by $\Phi(\alpha)=\alpha_c(\text{id}_c)$ as in the proof of the Yoneda Lemma. Just by unpacking the definitions of everything involved, naturality of $\Phi_{c,F}$ in $c$ is exactly that the following diagram commutes

$$\require{AMScd} \begin{CD} \mathcal{C}(c,c) @>\alpha_c>> Fc\\ @Vf_\ast VV @VVFfV\\ \mathcal{C}(c,d) @>>\alpha_d> Fd \end{CD} $$

for each natural transformation $\alpha: \mathcal{C}(c,-)\Rightarrow F$ and $f:c\to d$.

So naturality of this bijection is completely unavoidable. And we care about this specific bijection between $Fc$ and $\mathsf{Set}^\mathcal{C}(\mathcal{C}(c,-),F)$, as opposed to some other bijection which may not be natural, because when $F=\mathcal{C}(b,-)$ its inverse is the map on hom-sets $$よ_{b,c}:\mathcal{C}(b,c)\to \mathsf{Set}^\mathcal{C}(\mathcal{C}(c,-),\mathcal{C}(b,-))$$ associated the Yoneda embedding $よ: \mathcal{C}^\text{op}\to\mathsf{Set}^{\mathcal{C}}$. The bijection gives us full and faithfulness of $よ$, which in my experience is the most frequently used application of the Yoneda Lemma.

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