From another point of view, representable functors for a category C are the functors given with C. Their theory is a vast generalisation of upper sets in posets, and Yoneda's representability theorem generalizes Cayley's theorem in group theory.
A functor F : C → Set is said to be representable if it is naturally isomorphic to Hom(A,–) for some object A of C. A representation of F is a pair (A, Φ) where is a natural isomorphism.
A presheaf is representable when it is naturally isomorphic to the contravariant hom-functor Hom(–,A) for some object A of C. According to Yoneda's lemma, natural transformations from Hom(A,–) to F are in one-to-one correspondence with the elements of F(A).
In order to get a representation of F we want to know when the natural transformation induced by u is an isomorphism.
The Riesz representation theorem states that if F is continuous, then there exists a unique element
which represents F in the sense that F is equal to the inner product functional
For instance, the Dirac delta function is the distribution defined by
, and may be thought of as "represented" by an infinitely tall and thin bump function near
may be determined not by its values, but by its effect on other functions via the inner product.
Analogously, an object A in a category may be characterized not by its internal features, but by its functor of points, i.e. its relation to other objects via morphisms.
Stated in terms of universal elements: if (A1,u1) and (A2,u2) represent the same functor, then there exists a unique isomorphism φ : A1 → A2 such that Representable functors are naturally isomorphic to Hom functors and therefore share their properties.
In particular, (covariant) representable functors preserve all limits.
It follows that any functor which fails to preserve some limit is not representable.
Contravariant representable functors take colimits to limits.
Conversely, if K is represented by a pair (A, u) and all small copowers of A exist in C then K has a left adjoint F which sends each set I to the Ith copower of A.
Therefore, if C is a category with all small copowers, a functor K : C → Set is representable if and only if it has a left adjoint.
It follows that G has a left-adjoint F if and only if HomC(X,G–) is representable for all X in C. The natural isomorphism ΦX : HomD(FX,–) → HomC(X,G–) yields the adjointness; that is is a bijection for all X and Y.