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  "body": "\n## Summary\n\nTwo [[functors|28NQ0KRHD3]] $F \\colon \\mathcal{C} \\to \\mathcal{D}$ and $G \\colon \\mathcal{D} \\to \\mathcal{C}$ are adjoint when maps $F c \\to d$ match naturally with maps $c \\to G d$.\nThen $F$ is the left adjoint and $G$ the right adjoint.\nAdjunctions are everywhere in mathematics: free constructions, products, quotients, tensor and hom, and much more.\nEach adjunction gives a [[monad|FF6XXA6D1A]], and right adjoints preserve limits.\n\n## Explanations\n\n### L1 human\n\nMany constructions come with a forgetful partner.\nThe free group on a set $X$ is built from $X$, and every group has an underlying set.\nA group homomorphism from the free group on $X$ to a group $H$ is the same as a plain function from $X$ to the set of $H$.\nThis kind of matching between two sets of maps is an adjunction.\nHere the free group functor is the left adjoint and the forgetful functor is the right adjoint.\n",
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