In the definition of a category on Wikipedia, it is written that a category "consists of" a class of objects and a class of morphisms, as well as binary operations for compositions of morphisms.
What concerns me about this definition is that proper classes (such as the class of all sets or the class of all groups) are by definition not allowed to be elements of any other classes (or sets). Since the category of sets "consists of" the proper class of all sets, then if we take "consists of" to mean "contains as an element", it follows that this category must be a new type of collection that is larger than both classes and sets.
The only way I think that the category of sets could itself be a class would be if we take "consists of" to mean "contains as a subclass." However, I don't see anything on the page that clarifies this.
The following questions are related, although I don't think they address my specific issue about whether categories are larger than classes:
Best Answer
The subsequent is ugly, but it is a one way to avoid, if you want, some problems with proper classes in definition. We can define the category as some class $\mathcal{C}$ such that there exists class $\mathcal{A}$ of "arrows", for which $\mathcal{C}\subseteq\mathcal{A}\times\mathcal{A}\times\mathcal{A}$, and $(\beta,\alpha,\gamma)\in\mathcal{C}$ has a sense that arrows $\beta,\alpha$ can be composed, and $\beta\alpha=\gamma$. The corresponding axioms are obvious. Then $Ob(\mathcal{C})$ and hom-sets can be proper classes. Yet the set theory is narrow for category theory.