Universal Property - Formal Definition

Formal Definition

Suppose that U: D → C is a functor from a category D to a category C, and let X be an object of C. Consider the following dual (opposite) notions:

An initial morphism from X to U is an initial object in the category of morphisms from X to U. In other words, it consists of a pair (A, φ) where A is an object of D and φ: X → U(A) is a morphism in C, such that the following initial property is satisfied:

• Whenever Y is an object of D and f: X → U(Y) is a morphism in C, then there exists a unique morphism g: A → Y such that the following diagram commutes:

A terminal morphism from U to X is a terminal object in the comma category of morphisms from U to X. In other words, it consists of a pair (A, φ) where A is an object of D and φ: U(A) → X is a morphism in C, such that the following terminal property is satisfied:

• Whenever Y is an object of D and f: U(Y) → X is a morphism in C, then there exists a unique morphism g: Y → A such that the following diagram commutes:

The term universal morphism refers either to an initial morphism or a terminal morphism, and the term universal property refers either to an initial property or a terminal property. In each definition, the existence of the morphism g intuitively expresses the fact that (A, φ) is "general enough", while the uniqueness of the morphism ensures that (A, φ) is "not too general".