# The Unapologetic Mathematician

## Extraordinary Naturality

Now that we’ve gone back and rewritten the definition of naturality, let’s push it a bit.

First, notice that if we’re enriching over $\mathbf{Set}$ (in “ordinary” categories) then $\hom_\mathcal{D}(1_{S(A)},\eta_{B})$ means “take a morphism from $S(A)$ to $S(B)$ and follow it with $\eta_{B}$“. On the other hand, $\hom_\mathcal{D}(\eta_A,1_{T(B)})$ means “first do $\eta_A$, then follow it with a morphism from $T(A)$ to $T(B)$“. This recipe gives us back exactly the old naturality square, so $\mathbf{Set}$-natural transformations are exactly the ordinary natural transformations we’re familiar with!

So let’s take this reformulation of the naturality condition and tweak it. Instead of considering a family of arrows (in $\mathcal{D}_0$) $\eta_C:S(C)\rightarrow T(C)$, let’s move the variable over from the left to the right and consider a family $\beta_C:K\rightarrow T(C,C)$. Here, $K$ is an object of $\mathcal{D}$, and $T:\mathcal{C}^\mathrm{op}\otimes\mathcal{C}\rightarrow\mathcal{D}$ is a bifunctor. Now we say that the $\beta_C$ are the components of an “extraordinary $\mathcal{V}$-natural transformation” if the following diagram commutes:

This looks bizarre at first, though clearly it’s related to our revision of the enriched naturality diagram. It turns out that we’ve seen this sort of naturality before, though. If we read the diagram in $\mathbf{Set}$, consider a monoidal category $\mathcal{M}$ with duals, and use the functor $T(A,B)=A^*\otimes B$, then this is exactly the sort of naturality we find in the duality arrows $\eta_M:\mathbf{1}\rightarrow M^*\otimes M$!

Dually, we can define extraordinary $\mathcal{V}$-naturality for a family of morphisms $\gamma_C:T(C,C)\rightarrow K$. Write out this diagram, and show that the duality arrows $\eta_M:M\otimes M^*\rightarrow\mathbf{1}$ provide an example.

As another exercise, take these extraordinary naturality diagrams and work out the interpretation in $\mathbf{Set}$ explicitly. That is, actually start with some morphism $f:A\rightarrow B$ in the upper left-hand corner, and evaluate it all around. When we did this for our new $\mathcal{V}$-naturality diagram above we got our old naturality squares back. What do we get for extraordinary $\mathbf{Set}$-naturality?