Universal property 

In the sequel, we will give a general treatment of universal properties. It is advisable to study several examples first: product of groups and direct sum, free group, product topology, Stone-Čech compactification, tensor product, inverse limit and direct limit, kernel and cokernel[?], pullback[?], pushout[?] and equalizer[?].

Let C and D be categories, F : C -> D be a functor, and X an object of D. A universal morphism from F to X consists of an object A_X of C and a morphism φ_X : F(A_X) -> X in D, such that the following universal property is satisfied:

Whenever U is an object of C and φ : F(U) -> X is a morphism in D, then there exists a unique morphism ψ : U -> A_X such that φ_X F(ψ) = φ.

The existence of the morphism ψ intuitively expresses the fact that A_X is "large enough" or "general enough", while the uniqueness of the morphism ensures that A_X is "not too large".

From the definition, it follows directly that the pair (A_X, φ_X) is determined up to a unique isomorphism by X, in the following sense: if A'_X is another object of C and φ'_X : F(A'_X) -> X is another morphism which has the universal property, then there exists a unique isomorphism f : A_X -> A'_X such that φ'_X f = φ_X.

More generally, if φ_X1 : F(A_X1) -> X₁ and φ_X2 : F(A_X2) -> X₂ are two universal morphisms, and h : X₁ -> X₂ is a morphism in D, then there exists a unique morphism A_h: A_X1 -> A_X2 such that φ_X2 F(A_h) = φ_X1.

Therefore, if every object X of D admits a universal arrow, then the assignment X |-> A_X and h |-> A_h defines a covariant functor from D to C, the right-adjoint of F.

The dual concept of a co-universal construction also exists: it assigns to every object X of D an object B_X of C and a morphism ρ_X: X -> F(B_X) in D, such that the following universal property is satisfied:

Whenever U is an object of C and ρ : X -> F(U) is a morphism in D, then there exists a unique morphism σ : B_X -> U such that F(σ) ρ_X = ρ. A Co-universal constructions also defines a covariant functor from D to C, the so-called left-adjoint of F.

It is important to realize that not every functor F has a right-adjoint or a left adjoint; in other words: while one may always write down a universal property defining an object A_X, that doesn't mean that such an object also exists.

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