In my (limited) experience, it is usually easy to see when something is large enough to be a proper class, by constructing an element of the class for every set.

However, sometimes such a proper class has lots of redundant information, so we consider it modulo some equivalence relation. This way, in *some* cases, we end up with something small enough to be (represented by) **an actual set**!

Motivating example: The Witt group $W(F)$
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*(actually, we don't care about the group structure, I'm sorry)*

Let $F$ be a field. Let $$
 W(F) := \{(V, q)\ \text{quadratic}\ F\!-\!\text{vector spaces}\}/\sim
$$
Where $(V, q)\sim (W, r)$ if there exist metabolic (i.e., hyperbolic plus degenerate) spaces $E\_1, E\_2$ such that $(V, q)\perp E\_1 \simeq (W, r)\perp E\_2$, where $(V, q)\perp (W,r) = (V\oplus W, (v,w)\mapsto q(v)+r(w))$.

**Then $W(F)$ ~~is a set~~ has a set of representatives.**

When I first learned this, this was not at all obvious to me.

*(To be honest, I only vaguely recall this, and am not too sure of the technical details. The gist of this was however, that we only consider finite-dimensional vector spaces in the first place, which allow only for a finite number of non-equivalent quadratic forms, and even after that, most of those can be seen to be equivalent to a quadratic space of lower dimension by adding elements to make a large portion look hyperbolic)*

Question
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So, in a similar spirit to [this question](https://math.stackexchange.com/q/362446/91103):

> 
> What are other non-obvious examples of sets, e.g. proper classes being “quotiented” to a set?
> 
> 
>