Zhen Lin states one direction of the Yoneda Lemma: given $x\in F A$, the natural transformation $\theta_x:H_A\to F$ on $f:A\to B$ is $\theta(f)=F f\cdot x$. The other direction is that every $\theta:H_A\to F$ is of this form, where $x=\theta(\mathsf{id}_A)$. I leave it as an exercise to show that $\theta_x$ is natural and that this correspondence is bijective.

What I have just stated is the *natural mathematical argument* for the central idea of the Yoneda Lemma, albeit without the calculations required for the proofs. None of it has anything to do with Set Theory, whether that be ZFC, NBG or even NF.

This argument makes sense in a world in which objects and morphisms are things of a different sort from one another. Functors and natural transformations, on the other hand, are *schemes* or *recipes* that transform objects and morphisms into objects and morphisms. Functors and natural transformations are not themselves *things*.

We are quite used to this many-sorted language in algebra or first order logic. Even when the algebra itself has only one sort (if you don't know what I mean by that, I mean exactly the situation that you know), elements of the algebra, homomorphisms, predicates, ideals *etc.* belong to different sorts of the language in which we discuss them.

It was the innovation of Set Theory to *reify* schemes, *i.e.* turn them into *things*. In its unrestricted form, this innovation led to the famous *antinomies* or *paradoxes*, such as Russell's. The outcome of this was to develop restricted reification, such as Zermelo Set Theory.

Unfortunately, the mathematical community became addicted to Cantor's "paradise". So, even in disciplines such as Category Theory where we should have known better, we continue to find attempts at unrestricted reification such as Grothendieck Universes.

The disciplined alternative is the *type-theoretic style*, in which we nominate *certain* forms of reification, for example that of functions (*$\lambda$-abstraction*), and stick to just those (in the context of a particular discourse).

Now to reply to Michal Przybylek. I have just said and he agrees that "If people think that they need any kind of Set Theory to speak about Yoneda lemma, then they are wrong." However, far from being "philosophical" or "waving my hands", I have explained quite precisely what is the methodological set-up ("foundations" if you wish) behind the argument.

At least, I discussed the *ordinary form* of the argument. Of course, as mathematicians have always done extremely profitably with pre-existing arguments, one may re-interpret it in more exotic settings. Enriched, internal and locally cartesian closed categories are examples of such settings. Moreover they are also examples of desciplined restricted reification operations.

finitesets as the domain, in which case the ordinary Yoneda Lemma is enough. – Paul Taylor Nov 1 '13 at 15:20