On the Categories Without Uniqueness of cod and dom . Some Properties of the Morphisms and the Functors

In this paper C denotes a category and o1, o2, o3 denote objects of C. Let C be a non empty category structure with units and let o be an object of C. Observe that 〈o, o〉 is non empty. The following propositions are true: (1) Let v be a morphism from o1 to o2, u be a morphism from o1 to o3, and f be a morphism from o2 to o3. If u = f · v and f −1 · f = id(o2) and 〈o1, o2〉 6= ∅ and 〈o2, o3〉 6= ∅ and 〈o3, o2〉 6= ∅, then v = f −1 · u. (2) Let v be a morphism from o2 to o3, u be a morphism from o1 to o3, and f be a morphism from o1 to o2. If u = v · f and f · f −1 = id(o2) and 〈o1, o2〉 6= ∅ and 〈o2, o1〉 6= ∅ and 〈o2, o3〉 6= ∅, then v = u · f −1. (3) For every morphism m from o1 to o2 such that 〈o1, o2〉 6= ∅ and 〈o2, o1〉 6= ∅ and m is iso holds m−1 is iso. (4) For every non empty category structure C with units and for every object o of C holds ido is epi and mono.