A category is specified by four characteristics:

1. Objects, $X$

2. Morphisms, $X \to Y$. For every pair of objects in a category there exists a set whose elements map X to Y (this set could be called $Hom(X,Y)$)

3. Identitiy morphisms, a morphism $X \to X$

4. Composition. Given morphisms f and g, there exists a morphism h such that $h = f \circ g$

1. Unitality: for any morphism in the category $X \to Y$, $id_x \circ f  f  f \circ id_y#$
2. Associativity: given morphisms f, g, and h in the category, $(f \circ g) \circ h = f \circ (g \circ h)$