= Solution
A <monad> $\mathbb T=(T,\eta,\mu)$ 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
$$
\mu_A\eta_{TA}=1_{TA}=\mu_A T\eta_A,\qquad \mu_A\mu_{TA}=\mu_A T\mu_A.
$$
An <algebra for a monad> is $(A,a)$ with $a:TA\to A$ satisfying $a\eta_A=1_A$ and $a\mu_A=aT(a)$. A <morphism of algebras for a monad> $h:(A,a)\to(B,b)$ obeys $ha=bT(h)$. The <functor> laws show that identities and composites obey this equation, giving the <Eilenberg-Moore category> $\mathcal C^{\mathbb T}$.
For an <adjunction> $F\dashv G$, let $\eta$ be its <adjunction unit> and $\varepsilon$ its <adjunction counit>. The <monad induced by an adjunction> is
$$
\boxed{T=GF,\qquad \eta:1\Rightarrow GF,\qquad \mu_A=G(\varepsilon_{FA}).}
$$
The two unit laws are $G\varepsilon_{FA}\eta_{GFA}=1_{GFA}$ and $G(\varepsilon_{FA}F\eta_A)=1_{GFA}$, respectively the two <triangle identities for an adjunction>.
For associativity, <naturality> of the <adjunction counit> at $\varepsilon_{FA}:FGFA\to FA$ says
$$
\varepsilon_{FA}\,\varepsilon_{FGFA}=\varepsilon_{FA}\,FG(\varepsilon_{FA}).
$$
Apply $G$ to obtain $\mu_A\mu_{TA}=\mu_AT\mu_A$. The composites defining $\eta$ and $\mu$ are <natural transformations>, so all <monad> axioms have been checked, not only their underlying objectwise types.
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