**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".

Read more about this topic: Universal Property

### Other articles related to "formal definition":

... In typical usage, the

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### Famous quotes containing the words definition and/or formal:

“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the *definition* of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”

—Ralph Waldo Emerson (1803–1882)

“Two clergymen disputing whether ordination would be valid without the imposition of both hands, the more *formal* one said, “Do you think the Holy Dove could fly down with only one wing?””

—Horace Walpole (1717–1797)