linguistic:translation
Formal definition

To understand the definition of a universal construction, it is important to look at examples. Universal constructions were not defined out of thin air, but were rather defined after mathematicians began noticing a pattern in many mathematical constructions (see Examples below). Hence, the definition may not make sense to one at first, but will become clear when one reconciles it with concrete examples.

Let F: C → D {\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}} be a functor between categories C {\displaystyle {\mathcal {C}}} and D {\displaystyle {\mathcal {D}}}. In what follows, let X {\displaystyle X} be an object of D {\displaystyle {\mathcal {D}}}, A {\displaystyle A} and A ′ {\displaystyle A'} be objects of C {\displaystyle {\mathcal {C}}}, and h: A → A ′ {\displaystyle h:A\to A'} be a morphism in C {\displaystyle {\mathcal {C}}}.

Then, the functor F {\displaystyle F} maps A {\displaystyle A}, A ′ {\displaystyle A'} and h {\displaystyle h} in C {\displaystyle {\mathcal {C}}} to F (A) {\displaystyle F(A)}, F (A ′) {\displaystyle F(A')} and F (h) {\displaystyle F(h)} in D {\displaystyle {\mathcal {D}}}.

A universal morphism from X {\displaystyle X} to F {\displaystyle F} is a unique pair (A, u: X → F (A)) {\displaystyle (A,u:X\to F(A))} in D {\displaystyle {\mathcal {D}}} which has the following property, commonly referred to as a universal property:

For any morphism of the form f: X → F (A ′) {\displaystyle f:X\to F(A')} in D {\displaystyle {\mathcal {D}}}, there exists a unique morphism h: A → A ′ {\displaystyle h:A\to A'} in C {\displaystyle {\mathcal {C}}} such that the following diagram commutes:

We can dualize this categorical concept. A universal morphism from F {\displaystyle F} to X {\displaystyle X} is a unique pair (A, u: F (A) → X) {\displaystyle (A,u:F(A)\to X)} that satisfies the following universal property:

For any morphism of the form f: F (A ′) → X {\displaystyle f:F(A')\to X} in D {\displaystyle {\mathcal {D}}}, there exists a unique morphism h: A ′ → A {\displaystyle h:A'\to A} in C {\displaystyle {\mathcal {C}}} such that the following diagram commutes:

Note that in each definition, the arrows are reversed. Both definitions are necessary to describe universal constructions which appear in mathematics; but they also arise due to the inherent duality present in category theory.In either case, we say that the pair (A, u) {\displaystyle (A,u)} which behaves as above satisfies a universal property.