Editing Universal property (section)

===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.