# How does the Yoneda lemma imply that $\mathrm{Hom}(yC,P)$ is a set?

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?

• The functor $\mathrm{Hom}(-,C)$ is contravariant, it takes $(f:A\to B)$ to $(f\circ -):\mathrm{Hom}(B,C)\to \mathrm{Hom}(A,C)$ – Zev Chonoles Apr 8 '13 at 12:38
• 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. – Martin Brandenburg Apr 8 '13 at 12:42
• @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? – Zev Chonoles Apr 8 '13 at 12:44
• @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. – Zhen Lin Apr 8 '13 at 12:48
• @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? – Zev Chonoles Apr 8 '13 at 12:51

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.
• 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... – almaus Apr 8 '13 at 16:38