Size of a natural transformation and the Yoneda Lemma Appearing on the second page (under the section Digression: Size worries) of the following paper about the Yoneda Lemma: < It says that $a$ $priori$ $\mathcal C^{op},Set)$ is a class. I don't understand why this is the case. My understanding is that $a$ $priori$ each member of $\mathcal C^{op},Set)$ is a class. So it seems to me that $\mathcal C^{op},Set)$ 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

> So it seems to me that $\mathcal C^{op},Set$ 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 $\mathcal C^{op},X$ 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.