Coherence Condition - Associativity of Composition

Associativity of Composition

Let f : A → B, g : B → C and h : C → D be morphisms of a category containing objects A, B, C and D. By repeated composition, we can construct a morphism from A to D in two ways:

(h g) f : A → D, and

h (g f) : A → D.

We have now the following coherence statement:

(h g) f = h (g f).

In these two particular examples, the coherence statements are theorems for the case of an abstract category, since they follow directly from the axioms; in fact, they are axioms. For the case of a concrete mathematical structure, they can be viewed as conditions, namely as requirements for the mathematical structure under consideration to be a concrete category, requirements that such a structure may meet or fail to meet.