Examples of natural transformations (part 1)

I think natural transformations in particular need lots of examples, so here they are. Lets start out with categories $\mathcal C$ and $\mathcal D$, and a functor $F:\mathcal C\to\mathcal D$. Then the identity transformation (which I’ll write as $\mbox{id}$ is a natural transformation:

We need to check that we can make $\mbox{id}_X$ for each $X\in Ob(\mathcal C)$. such that

commutes. Note that by $\mbox{id}_X$ , I don’t mean the identity morphism on $X$. It has to be a morphism from $F(X)$ to $F(X)$. Let’s take $\mbox{id}_X=1_{F(X)}$. By $1_{F(X)}$, I mean the identity morphism on $F(X)$. Sorry for the notational bullshit. I know it’s annoying. I think this is the best way to do it, even though it’s particularly awful.

Going across the top and then down gives us $F(f)\circ\mbox{id}_X=F(f)\circ 1_{F(X)}=F(f)$. Going down and then across gives $\mbox{id}_Y\circ F(f)=1_{F(Y)}\circ F(f)=F(f)$. The diagram commutes, so the transformation is natural.

In terms of what this represents to us as mathematicians, it says that we can change a functor into itself by not doing anything. Moreover, this change is “natural” in the sense that it commutes in the way you would expect. Okay, maybe you wouldn’t expect it, but a category theorist would.