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I'm reading Awodey's Category Theory (1st ed) and at page 166 I did not found out the proof of one of the remarks (remark 8.4):

If $C$ is locally small, then $\mathsf{Sets}^{C^{\text{op}}}$ needs not be locally small. In this case, the Yoneda Lemma tells us that $\mathrm{Hom}(yC,P)$ is always a set.

I understand well what is the Yoneda lemma, and its proof, but I didn't understand why it could tell that $\mathrm{Hom}(yC,P)$ is a set... Note: here, $yC$ is the covariant Yoneda embedding applied to $C$, that is $\mathrm{Hom}(-,C)$.

Could anyone help me with some track?

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    $\begingroup$ The functor $\mathrm{Hom}(-,C)$ is contravariant, it takes $(f:A\to B)$ to $(f\circ -):\mathrm{Hom}(B,C)\to \mathrm{Hom}(A,C)$ $\endgroup$ – Zev Chonoles Apr 8 '13 at 12:38
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    $\begingroup$ Actually it is not a set, but it is isomorphic to a set. Since the isomorphism is canonical one can use it as an identification, so one can safely assume that it is a set. $\endgroup$ – Martin Brandenburg Apr 8 '13 at 12:42
  • $\begingroup$ @Martin: I'm not sure I understand the distinction (I know practically no set theory, unfortunately) - shouldn't providing a bijection from a class to a set demonstrate that the class is itself a set, and not a proper class, regardless of canonical-ness? $\endgroup$ – Zev Chonoles Apr 8 '13 at 12:44
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    $\begingroup$ @Zev It depends on the precise details of how $\textrm{Hom}(y C, P)$ is coded, but Martin's claim is correct in any sensible coding where natural transformations are coded as something that contains their graph. The bottom line is, a set is something that is hereditarily small. For example, if $U$ is the universe of all sets, then the collection $\{ U \}$ is not a set because one of its members is not a set. $\endgroup$ – Zhen Lin Apr 8 '13 at 12:48
  • $\begingroup$ @Zhen: Ah, I see - thanks for the explanation. But now I'm not sure why my answer is right... how does the canonical-ness help? $\endgroup$ – Zev Chonoles Apr 8 '13 at 12:51
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The Yoneda lemma tells you that the collection of natural transformations from the representable functor $\mathrm{Hom}(-,C):\mathcal{C}\to\mathsf{Set}$ to a contravariant functor $P:\mathcal{C}\to\mathsf{Set}$ is in bijection with the elements of $P(C)$, and thus is essentially small (not technically the same thing as being a set; see Martin's and Zhen's comments).

In the category $\mathsf{Set}^{\mathcal{C}^{\text{op}}}$, $\mathrm{Hom}(yC,P)$ is precisely this collection of natural transformations.

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  • $\begingroup$ For the reasons Martin has highlighted, it may be better to say "is small" or "is in bijection with a set" instead of "is a set". $\endgroup$ – Zhen Lin Apr 8 '13 at 12:50
  • $\begingroup$ I would say "essentially small". $\endgroup$ – Martin Brandenburg Apr 8 '13 at 12:59
  • $\begingroup$ In fact there are lots of essentially small classes appearing in mathematics. They appear as isomorphism classes of objects in essentially small categories. For example, vector bundles, f.g. projective modules, f.g. modules, finite-dimensional representations of a group, etc. One then usually just pretends that they are sets, and this works. $\endgroup$ – Martin Brandenburg Apr 8 '13 at 15:09
  • $\begingroup$ Many thanks!! In fact... I just forgot that in the statement of Yoneda $P$ is not "any" functor from $C$, but precisely a functor $P : C \rightarrow Set$ so that $PC$ is a set. And it is isomorph to $Hom(yC,P)$ so finally $Hom(yC,P)$ is a set... $\endgroup$ – almaus Apr 8 '13 at 16:38

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