The coproduct is given by the disjoint union: given sets Ai where i ranges over some index set I, we construct the coproduct as the union of Ai×{i} (the cartesian product with i serves to ensure that all the components stay disjoint).
Set is thus a topos (and in particular cartesian closed and exact in the sense of Barr).
If C is an arbitrary category, the contravariant functors from C to Set are often an important object of study.
One way to resolve the problem is to work in a system that gives formal status to proper classes, such as NBG set theory.
Assuming this extra axiom, one can limit the objects of Set to the elements of a particular universe.
In one variation of this scheme, the class of sets is the union of the entire tower of Grothendieck universes.
Instead, theorems are expressed in terms of the category SetU whose objects are the elements of a sufficiently large Grothendieck universe U, and are then shown not to depend on the particular choice of U.
As a foundation for category theory, this approach is well matched to a system like Tarski–Grothendieck set theory in which one cannot reason directly about proper classes; its principal disadvantage is that a theorem can be true of all SetU but not of Set.