= Free-forgetful Eilenberg-Moore adjunction
{title2=$F^{\mathbb T}\dashv G^{\mathbb T}$}
For a <monad> $(T,\eta,\mu)$, the <free algebra functor> sends $X$ to $(TX,\mu_X)$ and is a <left adjoint> to the <forgetful functor> from the <Eilenberg-Moore category>. The transpose <bijection> sends an <monad algebra morphism> $h:TX\to A$ to $h\eta_X$; its inverse sends $k:X\to A$ to $aT(k)$, where $a:TA\to A$ is the algebra structure. The <monad> and algebra laws make these inverse and natural. The <adjunction unit> is $\eta$ and the <adjunction counit> at $(A,a)$ is $a$, so the induced <monad> has exactly the original endofunctor, unit and multiplication.
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