Examples of Category Structures

In this paper A denotes a non empty set, S denotes a non void non empty many sorted signature, and x denotes a set. Let us consider A. The functor MSSCat(A) yields a strict non empty category structure and is defined by the conditions (Def. 1). (Def. 1) (i) The carrier of MSSCat(A) = MSS-set(A), (ii) for all elements i, j of MSS-set(A) holds (the arrows of MSSCat(A))(i, j) = MSS-morph(i, j), and (iii) for all objects i, j, k of MSSCat(A) such that i ∈ MSS-set(A) and j ∈ MSS-set(A) and k ∈ MSS-set(A) and for all functions f1, f2, g1, g2 such that 〈f1, f2〉 ∈ (the arrows of MSSCat(A))(i, j) and 〈g1, g2〉 ∈ (the arrows of MSSCat(A))(j, k) holds (the composition of MSSCat(A))(i, j, k)(〈g1, g2〉, 〈f1, f2〉) = 〈g1 · f1, g2 · f2〉. Let us consider A. Note that MSSCat(A) is transitive and associative and has units. The following proposition is true