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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?

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    For some related examples in posts here, see the [floor function](http://math.stackexchange.com/a/147832/242) and [gcds and lcms.](http://math.stackexchange.com/q/147900/242)2012-05-24

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This is explained in my blog post on the Yoneda lemma.

By the way, I do not consider this argument "nuking mosquitos." The Yoneda lemma is hardly a nuke; I would reserve that term for a highly technical result which requires a long proof. The proof of the Yoneda lemma is extraordinarily short and elegant. Besides, even this seemingly trivial special case can be surprisingly useful.

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    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.2012-05-25