= Solution
Write $\eta:1_{\mathcal C}\to GF$ and $\varepsilon:FG\to1_{\mathcal D}$ for the <unit and counit of an adjunction>. The induced <monad> has
$$
T=GF,\qquad \mu=G\varepsilon F:T^2\to T.
$$
The <Eilenberg-Moore comparison functor> sends $D\in\mathcal D$ to the <algebra for a monad>
$$
K(D)=(GD,G\varepsilon_D),
$$
and sends $v:D\to D'$ to $Gv$. The <triangle identities for an adjunction> give the algebra unit law, and naturality of the counit gives the algebra multiplication law and the algebra-morphism equation for $Gv$.
If $\mathcal D$ has <coequalizers>, define the <Left adjoint to the Eilenberg-Moore comparison functor> at an algebra $(A,a:TA\to A)$ by
$$
FTA\mathrel{\substack{\xrightarrow{Fa}\\[-2pt]\xrightarrow[\varepsilon_{FA}]{} }}FA\xrightarrow{q}L(A,a).
$$
The parallel pair is reflexive through $F\eta_A$. A <morphism> $L(A,a)\to D$ is equivalently an arrow $b:FA\to D$ with $bFa=b\varepsilon_{FA}$. Under $F\dashv G$, its transpose is $\bar b=Gb\,\eta_A:A\to GD$. The transpose of $bFa$ is $\bar b\,a$, while the transpose of $b\varepsilon_{FA}$ is $G\varepsilon_D\,T\bar b$. Hence the coequalizer condition is exactly
$$
\bar b\,a=G\varepsilon_D\,T\bar b,
$$
the equation for a <morphism of algebras for a monad> $(A,a)\to K(D)$. This supplies the natural bijection proving \b[$L\dashv K$], and also defines $L$ on algebra morphisms by the coequalizer universal property.
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