Examples of category structures

[10], and [11] provide the notation and terminology for this paper. The following proposition is true 1. PRELIMINARIES (1) For all sets X1, X2 and for all sets a1, a2 holds [:X1 ↦− → a1, X2 ↦− → a2:] = [:X1, X2:] ↦− → 〈a1, a2〉. Let I be a set. One can check that 0I is function yielding. Next we state two propositions: (2) For all functions f, g holds �(g · f) = g · � f. (3) For all functions f, g, h holds � ( f · [:g, h:]) = � f · [:h, g:]. Let f be a function yielding function. Observe that � f is function yielding. Next we state the proposition (4) Let I be a set and A, B, C be many sorted sets indexed by I. Suppose A is transformable to B. Let F be a many sorted function from A into B and G be a many sorted function from B into C. Then G ◦ F is a many sorted function from A into C. Let I be a set and let A be a many sorted set indexed by [:I, I:]. Then �A is a many sorted se