Concrete categories

for this paper. Let x be a set. The functor x1,1 yields a set and is defined by: (Def. 1) x1,1 = (x1)1. The functor x1,2 yields a set and is defined as follows: (Def. 2) x1,2 = (x1)2. The functor x2,1 yielding a set is defined by: (Def. 3) x2,1 = (x2)1. The functor x2,2 yielding a set is defined by: (Def. 4) x2,2 = (x2)2. In the sequel x, x1, x2, y, y1, y2 denote sets. The following proposition is true (1) 〈〈x1, x2〉, y 〉 1,1 = x1 and 〈〈x1, x2〉, y 〉 1,2 = x2 and 〈x, 〈y1, y2〉 〉 2,1 = y1 and 〈x, 〈y1, y2〉 〉 2,2 = y2. Let D1, D2, D3 be non empty sets and let x be an element of [:[:D1, D2:], D3:]. Then x1,1 is an element of D1. Then x1,2 is an element of D2. Let D1, D2, D3 be non empty sets and let x be an element of [:D1, [:D2, D3:]:]. Then x2,1 is an element of D2. Then x2,2 is an element of D3. For simplicity, we adopt the following convention: C, D, E are categories, c, c1 are objects of C, d, d1 are objects of D, x is a set, f, f1 are morphisms of E, g is a morphism of C, h is a morphism of D, F is a functor from C to E, and G is a functor from D to E