Editing Universal property (section)

==Connection with comma categories==
Universal morphisms can be described more concisely as initial and terminal objects in a [[comma category]] (i.e. one where morphisms are seen as objects in their own right).

Let <math>F: \mathcal{C} \to \mathcal{D}</math> be a functor and <math>X</math> an object of <math>\mathcal{D}</math>. Then recall that the comma category <math>(X \downarrow F)</math> is the category where 
* Objects are pairs of the form <math>(B, f: X \to F(B))</math>, where <math>B</math> is an object in <math>\mathcal{C}</math>
* A morphism from <math>(B, f: X \to F(B))</math> to <math>(B', f': X \to F(B'))</math> is given by a morphism <math>h: B \to B'</math> in <math>\mathcal{C}</math> such that the diagram commutes:
[[File:Definition of a morphism in a comma category.svg|center|A morphism in the comma category is given by the morphism <math>h: B \to B'</math> which also makes the diagram commute.]]
Now suppose that the object <math>(A, u: X \to F(A))</math> in <math>(X \downarrow F)</math> is initial. Then
for every object <math>(A', f: X \to F(A'))</math>, there exists a unique morphism <math>h: A \to A'</math> such that the following diagram commutes. 
[[File:Connection between universal diagrams and comma categories.svg|center|This demonstrates the connection between a universal diagram being an initial object in a comma category.]]

Note that the equality here simply means the diagrams are the same. Also note that the diagram on the right side of the equality is the exact same as the one offered in defining a '''universal morphism from <math>X</math> to <math>F</math>'''. Therefore, we see that a universal morphism from <math>X</math> to <math>F</math> is equivalent to an initial object in the comma category <math>(X \downarrow F)</math>.

Conversely, recall that the comma category <math>(F \downarrow X)</math> is the category where 
*Objects are pairs of the form <math>(B, f: F(B) \to X)</math> where <math>B</math> is an object in <math>\mathcal{C}</math>
*A morphism from <math>(B, f:F(B) \to X)</math> to <math>(B', f':F(B') \to X) </math> is given by a morphism <math>h: B \to B'</math> in <math>\mathcal{C}</math> such that the diagram commutes:
[[File:Definition of a morphism in a comma category 1.svg|center|This simply demonstrates the definition of a morphism in a comma category.]]
 
Suppose <math>(A, u:F(A) \to X) </math> is a terminal object in <math>(F \downarrow X)</math>. Then for every object <math>(A', f: F(A') \to X) </math>, 
there exists a unique morphism <math>h: A' \to A </math> such that the following diagrams commute. 
[[File:Connection between comma category and universal properties.svg|center|This shows that a terminal object in a specific comma category corresponds to a universal morphism.]]

The diagram on the right side of the equality is the same diagram pictured when defining a '''universal morphism from <math>F</math> to <math>X</math>'''. Hence, a universal morphism from <math>F</math> to <math>X</math> corresponds with a terminal object in the comma category 
<math>(F \downarrow X)</math>.