Usually, a collection of operations forms a set. But I've heard of an example where it is (must be) a proper class. Namely, that
$\mathbb{CompHauss}$
becomes a variety if proper class of operations is allowed. What are the operations? I have heard of ultrafilter convergences, but I do not know what they technically are. Why small set of them doesn't yield that $\mathbb{CompHauss}$ is a variety?
BTW, how the smallness of the collection of operations is used in the proof of Birkhoff's type theorem on $HSP$ being the variety?