If
- $F$ and $G$ are injective functors between the categories $C$ and $D$
- $H$ is an endofunctor on the category $D$ such that $H∘F = G$
- $η$ is a natural transformation from $F$ to $G$
Then for every morphism $f: x→y$ in the category $C$, we have:
$H((F f)(F x)) = H(F y) = G y = (G f)(G x) = (G f)(H (F x))$
After defining the morphisms
- $H_x: F x→G x$ defined as $H_x = H(F x)$
- $H_y: F y→G y$ defined as $H_y = H(F y)$
we get: $H_y∘F(f) = G(f)∘H_x$
Which looks the same as the commute rule of the natural transformation:
$η_y∘F(f) = G(f)∘η_x$
Is there an isomorphism between H and η?
I understand that in the general case we could not define a functor H for every natural transformation η, since the image of a functor doesn't have to be a category, but could we define a natural transformation η for every functor H which composes $H∘F = G$?
The only difference I see is that a natural transformation can be defined on "almost categories" which are just missing some compositions arrows, where as a functor is defined on categories, so is a functor just a special case of a natural transformation?
What's the importance on defining the natural transformation on functors vs on these "almost categories"?
Can you please help me understand what am I missing?
Firstly I might say there is no need to consider injective $F,G$.
The equation: $$H(Ff)(Fx)=(Gf)(H(Fx))$$ Isn't quite a naturality equation, as in: $$\eta_y\circ Ff=Gf\circ\eta_x$$
For a few reasons. Firstly, in the first equation $H$ is only ever playing the role of an object, either $H(Fx)=Gx$ or $H(Fy)=Gy$, whereas $\eta_\bullet$ is an arrow, not an object.
Secondly, while $\eta$ runs $F(-)\to G(-)$, even if you do try to interpret $H$ as an arrow in the above, as in $H(Ff):H(Fx)\to H(Fy)$, this is just an arrow $Gx\to Gy$. It is even just $Gf$ itself. It is not linking $Fx\to Gx$, and it is not linking $Ff$ with $Gf$.
Functors also act on things: they associate objects to objects and arrows to arrows. Natural transformations are just bags of arrows, and act neither on objects or arrows themselves.
Finally, you could define a natural transformation $F\implies G$ given $G=HF$ if you had a natural transformation $\mathrm{Id}\implies H$. Think about what the problem is: you need to find arrows $Fx\to HFx$ for all $x$. There's no reason for these arrows to exist, nevermind the question of existence of a natural transformation, unless you already have a natural association $z\to Hz$.
Functors play very different roles to natural transformations.