Universal property
Existence and uniqueness

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

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