Figure 1.

Requirements of a category. (a) Composition: if A, B, and C are objects in category X, and f: A → B and g: B → C are arrows in category X, then we can compose (or combine) f and g to obtain an arrow, f;g: A → C. Note f;g reads as “f then g” and it is often denoted as g°f (Fong and Spivak 2019). (b) Associativity: if f, g, and h are arrows in category X, then the order to compose the arrows does not matter: (f;g);h = f;(g;h). A, B, C, and D are objects in category X. (c) Unit: For any object A in category X, there is a self-referential arrow A → A, which is called identity arrow: 1A. For any arrow f: A → B, the following is always satisfied: 1A;f = f = f;1B