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

Write the adjunction unit as $\eta :1_{\mathcal{C}}\Rightarrow GF$ and the adjunction counit as $\epsilon :FG\Rightarrow 1_{\mathcal{D}}$. Define the monad induced by an adjunction bywith unit $\eta$. Naturality of $\epsilon$ makes $\mu$ a natural transformation.

The triangle identities for an adjunction give the two unit laws:For associativity, apply naturality of $\epsilon$ to the arrow ${\epsilon }_{FA}:FGFA\to FA$ in $\mathcal{D}$:Applying $G$ yieldsThus the induced data satisfy all the monad laws, rather than merely supplying an endofunctor and two transformations.