How can I have a collection of arrows which is not set in Category theory? I know some famous examples of large categories, where the set of objects is a proper class. How about a collection of arrows? Please give me examples if you can.
 A: Consider the category with a single object $*$ and an arrow $\alpha\colon *\to *$ for every ordinal $\alpha$. Define $\alpha\circ \beta = \max(\alpha,\beta)$, so that $0$ is the identity and associativity obviously holds. Since there are a proper class of ordinals, $\text{Hom}(*,*)$ is a proper class.
The point here is that a category with one object is a monoid, so to get a category with $\text{Hom}(*,*)$ large, we just need a large (proper class sized) monoid, and the ordinals do nicely.
For an even more trivial example, let $\mathbb{X}$ be any proper class, and consider the category with two objects $1$ and $2$, $\mathrm{Hom}(1,1) = {\text{id}_1}$, $\mathrm{Hom}(2,2) = {\text{id}_2}$, and $\mathrm{Hom}(1,2) = \mathbb{X}$. There is a unique way to define composition to make this into a category.
A: All of the standard large categories have a proper class of arrows. Every category has at least as many arrows as objects, since every object has an identity arrow. If you don't want a proper class of objects, then you're stuck with relatively unnatural examples, as in the other answer.
