Proof that Every family { A i |i I} in the category of sets has a

Proof that Every family { A_i |i ϵ I} in the category of sets has a coproduct. U A_i, = { (a,i) ε ( U A_i)} X I |a ε A_i} with A; → U A_i given by a ↦ (a,i). U A_i, is called the disjoint union of the sets A,.]