Size of a natural transformation and the Yoneda Lemma

Question

Appearing on the second page (under the section Digression: Size worries) of the following paper about the Yoneda Lemma:

http://www.maths.ed.ac.uk/~tl/categories/yoneda.ps

It says that $a$ $priori$ $[\mathcal C^{op},Set](H_A,X))$ is a class. I don't understand why this is the case.

My understanding is that $a$ $priori$ each member of $[\mathcal C^{op},Set](H_A,X))$ is a class.

So it seems to me that $[\mathcal C^{op},Set](H_A,X))$ could be a collection of proper classes.

I've looked at this How does the Yoneda lemma imply that $\mathrm{Hom}(yC,P)$ is a set?, but it hasn't helped me much.

Any help is appreciated -Thanks

Answer 1

So it seems to me that $[\mathcal C^{op},Set](HA,X)$ could be a collection of proper classes.

You're right. Each natural transformation $\alpha\colon H_A\to X$ is a class and proper when $Ob(\mathcal C)$ is proper. Therefore $[\mathcal C^{op},X](H_A,X)$ is apriori a conglomerate, see p 15-16 in Joy of Cats, which is just a extension of the class concept, much as in the same way class was an extension of set. Yoneda's lemma shows that this conglomerate is a small conglomerate and for all practical purposes can be considered a set.