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Stream: learning: questions

Topic: Understanding Yoneda Lemma "naturalness in F"

Davi Sales Barreira (May 30 2022 at 16:59):

In the Yoneda Lemma, it's said that:

Given a functor $F:\mathbf{Set} \to \mathbf{Set}$, and a set $S$, there is an isomorphism

$$F(S) \cong Nat(y^S, F)$$

where $Nat$ is the set of natural transformations. Moreover, the equation above is natural in both $S$ and $F$.

Now, the part I'm having trouble understanding is this final "natural in $F$". I mean, in the functor category $[\mathbf{Set}, \mathbf{Set}]$, the natural transformations are morphisms, so what are the 'things' that work as "Natural Transformations" in this category?

Mike Shulman (May 30 2022 at 17:08):

It means that for any morphism $F\to G$, which is itself a natural transformation, an appropriate square commutes relating this isomorphism for $F$ to this isomorphism for $G$ along the maps $F(S) \to G(S)$ and ${\rm Nat}(y^S,F) \to {\rm Nat}(y^S,G)$ induced by the natural transformation $F\to G$.

Davi Sales Barreira (May 30 2022 at 17:56):

Thanks, @Mike Shulman . It's amazing how a different phrasing can lead to suddenly understanding a definition.

Mike Shulman (May 30 2022 at 18:00):

It is!