# Excessive use of the Yoneda lemma

In a MathOverflow thread on "nuking mosquitos", Andrej Bauer offered the following proof:

If two elements in a poset have the same lower bounds then they are equal by Yoneda lemma.

I understand that a poset can be considered to be a category with at most one arrow between any two objects, and I understand the statement of the Yoneda lemma, although I have little experience in using it. But I do not understand this proof. How does the Yoneda Lemma help?

-
The [silly] tag is... silly. –  Asaf Karagila May 24 '12 at 20:59
For some related examples in posts here, see the floor function and gcds and lcms. –  Bill Dubuque May 24 '12 at 21:23

The nuke isn’t specifically the Yoneda lemma: it’s employing the wholly unnecessary language and machinery of category theory to prove a trivial result that follows immediately from the definition (each is a lower bound of the other, and the order is antisymmetric). Possibly it’s useful to see this fact as an instance of something more general $-$ I’m agnostic on that point $-$ and perhaps the machinery is something less than a nuke, but it’s certainly using a bulldozer to move a chickpea. –  Brian M. Scott May 25 '12 at 6:30