Size of a natural transformation and the Yoneda Lemma
Solution 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.