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

A monad on $\mathcal{C}$ consists of an endofunctor $T:\mathcal{C}\to \mathcal{C}$ and natural transformations $\eta :1_{\mathcal{C}}\Rightarrow T$ and $\mu :T^2\Rightarrow T$ satisfying, at every object $A$,These are the unit and associativity laws for the unit and multiplication of a monad.

An algebra for a monad is a pair $(A,a)$ with $a:TA\to A$ satisfyingA monad algebra morphism $f:(A,a)\to (B,b)$ is a morphism $f:A\to B$ with $fa=bTf$. Identities satisfy this condition, and composing two such maps preserves it by functoriality of $T$. The resulting Eilenberg-Moore category is denoted ${\mathcal{C}}^T$. Its forgetful functor $U:{\mathcal{C}}^T\to \mathcal{C}$ sends $(A,a)$ to $A$ and leaves the underlying arrows unchanged.