= Solution
Write the <adjunction unit> and <adjunction counit> of $F\dashv G$ as $\eta,\varepsilon$. Its induced <monad> is $T=GF$, with multiplication $\mu=G\varepsilon_F$. The <Eilenberg-Moore comparison functor> is
$$
K(d)=(Gd,a_d),\qquad a_d=G\varepsilon_d:T(Gd)\to Gd,\qquad K(h)=Gh.
$$
The unit law for $a_d$ is $G\varepsilon_d\eta_{Gd}=1_{Gd}$, a <triangle identity for an adjunction>. <Naturality> of $\varepsilon$ at $\varepsilon_d$ gives $\varepsilon_dFG\varepsilon_d=\varepsilon_d\varepsilon_{FGd}$; applying $G$ gives $a_dTa_d=a_d\mu_{Gd}$. <Naturality> at $h$ gives the <monad algebra morphism> equation for $Gh$. Thus $K$ is well-defined.
For each $d$, consider the fork in $\mathcal D$
$$
FGFGd\ \substack{\xrightarrow{\varepsilon_{FGd}}\\[-2pt]\xrightarrow[FG\varepsilon_d]{}}\ FGd\xrightarrow{\varepsilon_d}d.
$$
It commutes by the <naturality> equation just used. Its image under $G$ is the fork $T^2(Gd)\rightrightarrows T(Gd)\to Gd$ of the preceding question, with parallel <morphisms> $\mu_{Gd},Ta_d$ and final map $a_d$. It is split using $s=\eta_{Gd}$ and $t=\eta_{T(Gd)}$. By the assumed reflection, \b[$\varepsilon_d$ is the <coequalizer> of this pair in $\mathcal D$].
For faithfulness, if $Gh=Gk$ for $h,k:d\to e$, <naturality> gives $h\varepsilon_d=\varepsilon_eFGh=\varepsilon_eFGk=k\varepsilon_d$. A <coequalizer> is an <epimorphism>, so $h=k$.
For fullness, let $u:K(d)\to K(e)$ be a <monad algebra morphism>. Thus $u:Gd\to Ge$ satisfies $u a_d=a_eTu$. <Set> $v=\varepsilon_eF u:FGd\to e$. We verify explicitly that $v$ equalizes the two <morphisms> of the displayed fork. The <monad algebra morphism> equation gives
$$
vFG\varepsilon_d=\varepsilon_eF(uG\varepsilon_d)
=\varepsilon_eFG\varepsilon_e\,FGF u.
$$
<Naturality> of $\varepsilon$ first at $Fu$ and then at $\varepsilon_e$ gives
$$
v\varepsilon_{FGd}=\varepsilon_e\varepsilon_{FGe}\,FGF u
=\varepsilon_eFG\varepsilon_e\,FGF u.
$$
The <coequalizer> therefore supplies a unique $h:d\to e$ with $h\varepsilon_d=v$. Applying $G$ yields $Gh\,a_d=Gv=a_eTu=u a_d$. The map $a_d$ is split epic with section $\eta_{Gd}$, so cancellation gives $Gh=u$. This proves fullness, and the earlier cancellation proves faithfulness. Hence
$$
\boxed{K:\mathcal D\to\mathcal C^T\text{ is full and faithful}.}
$$
No claim that $K$ is essentially surjective is needed; that stronger conclusion needs the corresponding existence hypotheses.
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