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  "body": "\n## Statement\n\nLet $F \\colon \\mathcal{C} \\to \\mathcal{D}$ be a functor.\n\n1. If $G$ and $G'$ from $\\mathcal{D}$ to $\\mathcal{C}$ are both [[right adjoint|TZHF6ZM92S]] to $F$, then there is a natural isomorphism $G \\cong G'$.\n2. Dually, if $F$ and $F'$ from $\\mathcal{C}$ to $\\mathcal{D}$ are both left adjoint to a functor $G$, then $F \\cong F'$.\n\n## Remarks\n\nFor each $d$, the adjunctions give bijections $\\Hom(c, G d) \\cong \\Hom(F c, d) \\cong \\Hom(c, G' d)$, natural in $c$.\nBy the Yoneda lemma, these come from a unique isomorphism $G d \\cong G' d$.\nNaturality in $d$ makes these isomorphisms a [[natural isomorphism|KYCPCC57FT]].\nThe isomorphism is the unique one compatible with the two adjunctions.\n\nExample: the free group on a set can be built from reduced words or as a quotient of a free monoid.\nBoth constructions are left adjoint to the forgetful functor, so they give isomorphic functors.\nLikewise the abelianization $H \\mapsto H/[H, H]$ is determined up to isomorphism by being left adjoint to the inclusion of abelian groups into groups.\n\n## Explanations\n\n### L1 human\n\nAn adjoint of a functor, when it exists, is unique up to isomorphism.\nSo it does not matter which construction one uses for a free object or a quotient, as long as it has the right universal property.\nThis is why one may speak of the left adjoint of a functor.\n\n### L2 human\n\nThe proposition is a form of the uniqueness of objects with a universal property, applied object by object.\nIt lets one define a functor by asking that it be an adjoint, as for the tensor product, the abelianization or the Stone–Čech compactification.\nIt also shows that adjunctions compose up to isomorphism: if $F \\dashv G$ and $F' \\dashv G'$, then $F' F \\dashv G G'$, and any other left adjoint of $G G'$ is isomorphic to $F' F$.\n",
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