Categories without Uniqueness of cod and dom

where ◦o1,o2,o3 : 〈o2,o3〉×〈o1,o2〉 → 〈o1,o3〉 that satisfies usual conditions (associativity and the existence of the identities). This approach is closer to the way in which categories are presented in homological algebra (e.g. [1], pp.58-59). We do not assume that 〈o1,o2〉’s are mutually disjoint. If f is simultaneously a morphism from o1 to o2 and o1 to o2 (o1 6= o1) than different compositions are used (◦o1,o2,o3 or ◦o′1,o2,o3 ) to compose it with a morphism g from o2 to o3. The operation g · f has actually six arguments (two visible and four hidden: three objects and the category). We introduce some simple properties of categories. Perhaps more than necessary. It is partially caused by the formalization. The functional categories are characterized by the following properties: