Editing Universal property (section)

===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>