# Back to functors

I want to tie this all back to functors. We have a sort of “product functor” which takes objects $A$ in a category $\mathcal C$ and sends them to objects $A\times B$ in $\mathcal C$, for a prespecified $B$. As for morphisms, if $f:A\to A'$, then the product functor yields the map $f\times\mbox{id}_B:A\times B\to A'\times B$. You can check that this is really functorial.

I wanted to talk about products, mainly because they’re an important construction, but also because thinking of them in this context is good motivation for what we’re about to talk about… the natural transformations.

Natural transformations are confusing; there’s no doubt there. Particularly so, as I’m not telling you what they are yet. Hell, functors are confusing until you get a grasp on what they are, and the kind of magic they perform. Let me whet your appetite with this: We had morphisms, and then we built functors, which are maps on morphisms (and objects, but forget those for a moment). We have to go deeper! A natural transformation is the next layer down: They are maps on functors. BRRRRAAAAAAWWWRRRMRMM.