= Solution
A <monad> on $\mathcal C$ consists of an <endofunctor> $T$ and <natural transformations> $\eta:1\to T$, $\mu:T^2\to T$ satisfying, at every object $A$,
$$
\mu_A T\mu_A=\mu_A\mu_{TA},\qquad\mu_A\eta_{TA}=1_{TA}=\mu_A T\eta_A.
$$
These are the associativity and unit laws. An <algebra for a monad> is $(A,a:TA\to A)$ with
$$
a\eta_A=1_A,\qquad aT a=a\mu_A.
$$
A <monad algebra morphism> $h:(A,a)\to(B,b)$ is a <morphism> $h:A\to B$ satisfying $ha=bTh$. <Identity morphisms> satisfy this equation, and if $h,k$ satisfy it, then $kha=kbTh=cTkTh=cT(kh)$. Hence these objects and <morphisms> form the <Eilenberg-Moore category> $\mathcal C^T$, with the <forgetful functor> $U(A,a)=A$ and $Uh=h$.
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