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  "body": "\n## Summary\n\nA set-valued [[functor|28NQ0KRHD3]] is representable when it is naturally isomorphic to a [[hom functor|VJXS8AK67E]] $\\Hom(A, -)$ or $\\Hom(-, A)$ for some object $A$, called a representing object.\nRepresentability is the formal language of universal properties: products, kernels, free objects and many other constructions are representing objects of suitable functors.\nThe representing object is unique up to unique isomorphism.\n\n## Explanations\n\n### L1 human\n\nSome ways of producing sets from objects are secretly of the form \"all arrows from a fixed object $A$\".\nFor example, in the category of sets, consider the functor that sends a set $X$ to the set of its subsets.\nIt is the same as \"all functions from $X$ to the two-point set $\\{0, 1\\}$\", because a subset is described by its indicator function.\nSo the power set functor, viewed as contravariant, is represented by $\\{0, 1\\}$.\nRecognising a functor as representable tells you that one object and one special element carry all the information.\n",
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