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  "body": "\n## Definition\n\nLet $U \\colon \\mathcal{D} \\to \\mathcal{C}$ be a functor with a [[left adjoint|TZHF6ZM92S]] $F$ and counit $\\varepsilon \\colon F U \\Rightarrow \\mathrm{Id}_{\\mathcal{D}}$.\nLet $T = U F$ be the [[monad of the adjunction|DF5N0PK3ZS]], and let $\\mathcal{C}^T$ be its [[Eilenberg–Moore category|FEPRJR3MND]].\nThe *comparison functor* $K \\colon \\mathcal{D} \\to \\mathcal{C}^T$ sends an object $d$ to the algebra $(U(d), U(\\varepsilon_d))$ and a morphism $f$ to $U(f)$.\nThe functor $U$ is *monadic* if $K$ is an [[equivalence of categories|K7A04ZBD92]].\n\n## Remarks\n\nA functor without a left adjoint is never monadic.\nSome authors call $U$ strictly monadic when $K$ is an isomorphism of categories.\nThe comparison functor always satisfies $U^T \\circ K = U$, where $U^T$ is the forgetful functor of $\\mathcal{C}^T$.\n\nExample: the forgetful functor from groups to sets is monadic.\nIts left adjoint is the free group functor.\nAn algebra of the monad $T$ is a set $A$ with a map from the free group on $A$ to $A$ satisfying the algebra laws.\nSuch a map comes from a unique group structure on $A$, and algebra morphisms are group homomorphisms, so $K$ is an isomorphism.\n\nNon-example: the forgetful functor $U$ from topological spaces to sets has the discrete space functor as left adjoint.\nThe monad $U F$ is the identity monad, whose algebras are just sets, so $\\mathcal{C}^T$ is the category of sets.\nThe comparison functor sends a space to its underlying set, and it is not full, since not every function between spaces is continuous.\nSo $K$ is not an equivalence and $U$ is not monadic.\n",
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