= 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 $A$,
$$
\boxed{\mu_A T\eta_A=1_{TA}=\mu_A\eta_{TA},\qquad
\mu_A T\mu_A=\mu_A\mu_{TA}.}
$$
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 $a:TA\to A$ satisfying
$$
\boxed{a\eta_A=1_A,\qquad a\mu_A=aT a.}
$$
A <monad algebra morphism> $f:(A,a)\to(B,b)$ is a <morphism> $f:A\to B$ with $fa=bT f$. 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.
Back to article page
