= Solution
A <monad> on $\mathcal C$ consists of an endofunctor $T$ and <natural transformations> $\eta:1\Rightarrow T$, $\mu:T^2\Rightarrow T$, satisfying
$$
\mu\,T\eta=1_T=\mu\,\eta_T,\qquad\mu\,T\mu=\mu\,\mu_T.
$$
An <algebra for a monad> is $(A,a)$ with $a:TA\to A$, $a\eta_A=1_A$ and $a\mu_A=aT(a)$. A morphism $(A,a)\to(B,b)$ is an arrow $h:A\to B$ with $ha=bT(h)$. These form the <Eilenberg-Moore category> $\mathcal C^T$.
A <functor> $G:\mathcal D\to\mathcal C$ is monadic when it has a left adjoint $F$ and its comparison $K:\mathcal D\to\mathcal C^{GF}$ is an equivalence; this is the <monadic adjunction> convention. Some strict formulations require $K$ to be an <isomorphism> over the base. The distinction affects the meaning of literal limit creation, discussed in part (iii), but not the algebra construction.
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