Editing Universal property (section)

==Properties==

===Existence and uniqueness===

Defining a quantity does not guarantee its existence. Given a functor <math>F: \mathcal{C} \to \mathcal{D}</math> and an object <math>X</math> of <math>\mathcal{D}</math>, 
there may or may not exist a universal morphism from <math>X</math> to <math>F</math>.  If, however, a universal morphism <math>(A, u)</math> does exist, then it is essentially unique. 
Specifically, it is unique [[up to]] a ''unique'' [[isomorphism]]: if <math>(A', u')</math> is another pair, then there exists a unique isomorphism 
<math>k: A \to A'</math> such that <math>u' = F(k) \circ u</math>.
This is easily seen by substituting <math>(A, u')</math> in the definition of a universal morphism.

It is the pair <math>(A, u)</math> which is essentially unique in this fashion. The object <math>A</math> itself is only unique up to isomorphism. Indeed, if <math>(A, u)</math> is a universal morphism and <math>k: A \to A'</math> is any isomorphism then the pair <math>(A', u')</math>, where <math>u' = F(k) \circ u</math> is also a universal morphism.

===Equivalent formulations===

The definition of a universal morphism can be rephrased in a variety of ways. Let <math>F: \mathcal{C} \to \mathcal{D}</math> be a functor and let <math>X</math> be an object of <math>\mathcal{D}</math>. Then the following statements are equivalent:
* <math>(A, u)</math> is a universal morphism from <math>X</math> to <math>F</math>
* <math>(A, u)</math> is an [[initial object]] of the [[comma category]] <math>(X \downarrow F)</math>
* <math>(A, F(\bullet)\circ u)</math> is a [[representable functor|representation]] of <math>\text{Hom}_\mathcal{D}(X, F(-))</math>, where its components <math>(F(\bullet)\circ u)_B:\text{Hom}_{\mathcal{C}}(A, B) \to \text{Hom}_{\mathcal{D}}(X, F(B))</math> are defined by

<math display="block">(F(\bullet)\circ u)_B(f:A\to B):X\to F(B) = F(f)\circ u:X\to F(B)</math>

for each object <math>B</math> in <math>\mathcal{C}.</math>

The dual statements are also equivalent:
* <math>(A, u)</math> is a universal morphism from <math>F</math> to <math>X</math>
* <math>(A, u)</math> is a [[terminal object]] of the comma category <math>(F \downarrow X)</math>
* <math>(A, u\circ F(\bullet))</math> is a representation of <math>\text{Hom}_\mathcal{D}(F(-), X)</math>, where its components <math>(u\circ F(\bullet))_B:\text{Hom}_{\mathcal{C}}(B, A)\to \text{Hom}_{\mathcal{D}}(F(B), X)</math> are defined by

<math display="block"> (u\circ F(\bullet))_B(f:B\to A):F(B)\to X = u\circ F(f):F(B)\to X</math>

for each object <math>B</math> in <math>\mathcal{C}.</math>

===Relation to adjoint functors===

Suppose <math>(A_1, u_1)</math> is a universal morphism from <math>X_1</math> to <math>F</math> and <math>(A_2, u_2)</math> is a universal morphism from <math>X_2</math> to <math>F</math>. 
By the universal property of universal morphisms, given any morphism <math>h: X_1 \to X_2</math> there exists a unique morphism <math>g: A_1 \to A_2 </math> such that the following diagram commutes:
[[File:Connection between universal elements inducing a functor.svg|center|Universal morphisms can behave like a natural transformation between functors under suitable conditions.]]

If ''every'' object <math>X_i</math> of <math>\mathcal{D}</math> admits a universal morphism to <math>F</math>, then the assignment <math>X_i \mapsto A_i</math> and <math>h \mapsto g</math> defines a functor <math>G: \mathcal{D} \to \mathcal{C}</math>. The maps <math>u_i</math> then define a [[natural transformation]] from <math>1_\mathcal{D}</math> (the identity functor on <math>\mathcal{D}</math>) to <math>F\circ G</math>. The functors <math>(F, G)</math> are then a pair of [[adjoint functor]]s, with <math>G</math> left-adjoint to <math>F</math> and <math>F</math> right-adjoint to <math>G</math>.

Similar statements apply to the dual situation of terminal morphisms from <math>F</math>. If such morphisms exist for every <math>X</math> in <math>\mathcal{C}</math> one obtains a functor <math>G: \mathcal{C} \to \mathcal{D}</math> which is right-adjoint to <math>F</math> (so <math>F</math> is left-adjoint to <math>G</math>).

Indeed, all pairs of adjoint functors arise from universal constructions in this manner. Let <math>F</math> and <math>G</math> be a pair of adjoint functors with unit <math>\eta</math> and co-unit <math>\epsilon</math> 
(see the article on [[adjoint functors]] for the definitions). Then we have a universal morphism for each object in <math>\mathcal{C}</math> and <math>\mathcal{D}</math>:
*For each object <math>X</math> in <math>\mathcal{C}</math>, <math>(F(X), \eta_X)</math> is a universal morphism from <math>X</math> to <math>G</math>. That is, for all <math>f: X \to G(Y)</math> there exists a unique <math>g: F(X) \to Y</math> for which the following diagrams commute.
*For each object <math>Y</math> in <math>\mathcal{D}</math>, <math>(G(Y), \epsilon_Y)</math> is a universal morphism from <math>F</math> to <math>Y</math>. That is, for all <math>g: F(X) \to Y</math> there exists a unique <math>f: X \to G(Y)</math> for which the following diagrams commute.

[[File:Universal morphisms appear as the unit and counit of adjunctions.svg|center|The unit and counit of an adjunction, which are natural transformations between functors, are an important example of universal morphisms.]]

Universal constructions are more general than adjoint functor pairs: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of <math>\mathcal{C}</math> (equivalently, every object of <math>\mathcal{D}</math>).