= Solution
A <monad> on $\mathcal C$ consists of an <endofunctor> $T$ and <natural transformations> $\eta:1\to T$, $\mu:T^2\to T$ satisfying the <unit and multiplication of a monad> laws
$$
\boxed{\mu\circ T\eta=1_T=\mu\circ\eta T,\qquad
\mu\circ T\mu=\mu\circ\mu T.}
$$
An <algebra for a monad> is an object $A$ equipped with $a:TA\to A$ such that
$$
a\eta_A=1_A,\qquad aT(a)=a\mu_A.
$$
A <morphism of algebras for a monad> $f:(A,a)\to(B,b)$ is a <morphism> $f:A\to B$ satisfying $fa=bT(f)$. Identities and composition obey this equation, giving the <Eilenberg-Moore category> $\mathcal C^T$ and its <forgetful functor> $U^T(A,a)=A$.
For an <adjunction> $F\dashv U:\mathcal D\to\mathcal C$ with induced <monad> $T=UF$, the <Eilenberg-Moore comparison functor> is
$$
K:\mathcal D\to\mathcal C^T,\qquad
K(D)=(UD,U\varepsilon_D),\quad K(h)=Uh.
$$
The <functor> $U$ is monadic when it has such a <left adjoint> and this comparison is an <equivalence of categories>; a stricter convention requires an <isomorphism of categories> over $\mathcal C$. We will identify which limit-lifting conclusion each convention supports in part (iii).
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