Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 23 / 7 / c / Solution

Define the free algebra functor byThe monad laws say precisely that ${\mu }_X$ defines an monad algebra structure on $TX$, while naturality of $\mu$ makes $Tf$ an monad algebra morphism. We construct the adjunction explicitly.

For an monad algebra $(A,a)$ and a morphism $f:X\to A$, put $\overline{f}=aTf:TX\to A$. It is an monad algebra morphism becauseConversely, from an monad algebra morphism $h:(TX,{\mu }_X)\to (A,a)$ obtain $h{\eta }_X:X\to A$. These operations are inverse: naturality of $\eta$ and the monad algebra unit law giveand the algebra-morphism equation together with the monad unit law givesThe formulas commute with precomposition in $X$ and composition with monad algebra morphisms in $(A,a)$, so the bijections are natural. Therefore the free-forgetful Eilenberg-Moore adjunction isIts unit is ${\eta }_X$ and its counit at $(A,a)$ is the monad algebra action $a$.