# Yoneda’s Lemma (part 3)

Yesterday’s post probably made you woozy. It was long and tedious, and I didn’t even do all of the lemma. I skipped the parts about it being natural in the two functors. Eh, deal with it. It all comes down to checking that things commute. If you have the patience, and you go slowly enough, you can make it work, though I’m not sure what sort of insight it gives you.

So let’s summarize. Yoneda states that for locally small categories,

$\mbox{Nat}(h^A,F)\cong F(A)$,

and the proof can be summed up in the following diagram (source: Wikipedia entry on Yoneda’s Lemma)

Wikipedia’s diagram is slightly nice than mine, in that it tells you what the maps are right on the diagram. Very pretty. Good job, internet. What would we do without you?

Okay, great, so what can we do with this? There’s a nice simple corollary to Yoneda’s lemma that you may already know. We’ll show it next time.