Representable functor

Summary

A set-valued functor is representable when it is naturally isomorphic to a hom functor $\Hom(A, -)$ or $\Hom(-, A)$ for some object $A$, called a representing object. Representability is the formal language of universal properties: products, kernels, free objects and many other constructions are representing objects of suitable functors. The representing object is unique up to unique isomorphism.