In chapter 4 of Handbook of Categorical Algebra, vol 1, the author defines a "subobject of $A$" as "an equivalence class of monomorphisms with codomain $A$" (for a suitable notion of equivalence). He then defines what it means for a category to be well-powered: "$\mathcal{A}$ is well-powered when the subobjects of every object constitute a set". Thus, for instance, the category of sets is well-powered.
I'm having trouble understanding exactly what it means to have such a set of subobjects. As far as I can tell, each element of a set should also be a set, but an equivalence class of monomorphisms could be a proper class: for instance, the class of singleton sets is not a set. On the other hand, it seems that one can cheat by defining a subobject to be a class containing one representative of each equivalence class of monomorphisms, even though this is not, strictly speaking, what's stated in the book.
How can one solve this problem? Is there a "normal" set theory where such a set of subobjects can contain proper classes? Or does one need to cheat like suggested above?