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  "body": "\n## Statement\n\nLet $F \\colon \\mathcal{C} \\to \\mathcal{D}$ be [[left adjoint|TZHF6ZM92S]] to $G \\colon \\mathcal{D} \\to \\mathcal{C}$.\nLet $\\eta \\colon \\mathrm{Id}_{\\mathcal{C}} \\Rightarrow G F$ be the [[unit and $\\varepsilon \\colon F G \\Rightarrow \\mathrm{Id}_{\\mathcal{D}}$ the counit|MYGWT305AB]] of the adjunction.\nThen $(G F, \\eta, G \\varepsilon F)$ is a [[monad|FF6XXA6D1A]] on $\\mathcal{C}$.\nHere $G \\varepsilon F$ is the natural transformation $G F G F \\Rightarrow G F$ with components $G(\\varepsilon_{F(X)})$.\n\n## Remarks\n\nThe unit laws follow from the triangle identities $\\varepsilon F \\circ F \\eta = \\id_F$ and $G \\varepsilon \\circ \\eta G = \\id_G$.\nApplying $G$ to the first gives $G\\varepsilon F \\circ G F \\eta = \\id_{G F}$, and whiskering the second by $F$ gives $G \\varepsilon F \\circ \\eta G F = \\id_{G F}$.\nAssociativity reduces to $\\varepsilon \\circ F G \\varepsilon = \\varepsilon \\circ \\varepsilon F G$, which is the naturality of $\\varepsilon$.\n\nExample: the free monoid functor from sets to monoids is left adjoint to the forgetful functor.\nThe monad $G F$ sends a set $X$ to the set of finite words in $X$, and its multiplication concatenates a word of words.\nThis is the list monad.\n\nDually, $(F G, \\varepsilon, F \\eta G)$ is a [[comonad|97NPWM3S6E]] on $\\mathcal{D}$.\nConversely, every monad arises from some adjunction, for example from its Eilenberg–Moore category or its Kleisli category.\nDifferent adjunctions can give the same monad.\n\n## Explanations\n\n### L1 human\n\nWhen a free construction and a forgetful functor are adjoint, going free and then forgetting is a monad.\nIts unit puts an element into the free object as a generator.\nIts multiplication evaluates a free expression in free expressions as one free expression.\nThis is the main source of monads in practice.\n\n### L2 human\n\nThis result is half of the link between adjunctions and monads.\nThe other half says that every monad comes from an adjunction, and that among these there is an initial one, the Kleisli adjunction, and a terminal one, the Eilenberg–Moore adjunction.\nA right adjoint is called monadic when the comparison functor to the Eilenberg–Moore category is an equivalence.\nBeck's monadicity theorem tells when this happens.\nFor example, the free group adjunction gives a monad on sets whose algebras are exactly the groups.\n",
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