Are categories larger than classes?

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.