Are operations upon index-classes (rather than index-sets) allowed in mathematics? Why we always see extended operations, like arbitrary unions, products, etc. in different parts of mathematics in the form of extensions of finite ones upon arbitrary sets (called index set)? Why never use an index-class for a proper class?
EDIT:
Of course I think this a reason, for example, that in the context of category theory, some theorems only hold for small/locally small categories: we are not allowed to operate upon arbitrary proper classes.
 A: The restrictions on what you can use proper classes for are there to make sure that everything you write down can actually be proved to exist using the restricted set-building axioms of ZF set theory. We can prove once and for all that a union or product (or whatever) indexed by a set will always describe something that has an existence proof, but this proof generally doesn't go through if the indices are pulled from a proper class.
It's more of a rule of thumb than an absolute truth, though -- for example, an intersection indexed by a proper class is actually unproblematic from a ZF point of view. But it is much easier to remember simply not using proper classes for indexing, and the cases where you need the exceptions are sufficiently rare that it is cost-effective (in mental effort) simply to say not to index by proper classes, and in the few cases where that is insufficient argue directly from the axioms instead of relying on the slick index notation.
The reason why ZF restricts set-building in the first place is to avoid allowing the reasoning that leads to set-theoretical paradoxes such as Russel's or Buralli-Forti's paradox. There are a few competing suggestions for other restrictions that aim to achieve that by other means, but the general consensus is that the ZF restrictions are the most convenient to stay within for most ordinary mathematics.
A: The greatest operation that you will see is the power of the universe $V^A$ with $A$ set. This is a proper class, but the elements are sets: functions $f:A\longrightarrow V$. In the case $\text{set}^{\text{proper class}}$ the possible "elements" are proper classes.
