= Solution
Assume the <coequalizers> from part (b) exist and write them $q_a:FA\to L(A,a)$. The underlying arrow of the comparison-adjunction unit is
$$
\lambda_a=Gq_a\eta_A:A\to GL(A,a).
$$
We first relate it to the <monad algebra> action. <Naturality> of $\eta$, the <coequalizer> equation and a <monad> unit law give
$$
\begin{aligned}
\lambda_a a
&=Gq_a\eta_A a=Gq_a,Ta\,\eta_{TA}\\
&=G(q_aFa)\eta_{TA}=G(q_a\varepsilon_{FA})\eta_{TA}\\
&=Gq_a\mu_A\eta_{TA}=Gq_a.
\end{aligned}
$$
In $\mathcal C$, $a:TA\to A$ is itself a <split coequalizer> of $Ta,\mu_A:T^2A\rightrightarrows TA$. Indeed,
$$
aTa=a\mu_A,\quad a\eta_A=1_A,\quad
\mu_A\eta_{TA}=1_{TA},\quad Ta\eta_{TA}=\eta_Aa.
$$
For an explicit universal-property proof, if $r:TA\to X$ satisfies $rTa=r\mu_A$, then $s=r\eta_A$ obeys
$$
sa=r\eta_Aa=rTa\eta_{TA}=r\mu_A\eta_{TA}=r.
$$
Its uniqueness follows by composing any equation $sa=r$ with $\eta_A$.
Now $Gq_a$ equalizes $Ta$ and $\mu_A$. If it is a <coequalizer>, then $a$ and $Gq_a$ are <coequalizers> of the same pair. Their unique comparison map is $\lambda_a$, since $\lambda_a a=Gq_a$, and is therefore invertible. Conversely, if $\lambda_a$ is invertible, $Gq_a=\lambda_a a$ is an isomorphic copy of the <coequalizer> $a$, so it is a <coequalizer> too.
The <forgetful functor> $\mathcal C^T\to\mathcal C$ reflects <isomorphisms>: if an <monad algebra> <morphism> $u$ has an inverse as an underlying arrow, the equation $ua=bTu$ rearranges to $u^{-1}b=aT(u^{-1})$, making that inverse an <monad algebra> <morphism>. Thus underlying invertibility here is exactly invertibility of the unit component. For all <monad algebras> together, the <comparison-adjunction unit criterion> is
$$
\boxed{\lambda\text{ is invertible}\quad\Longleftrightarrow\quad
G\text{ preserves every specified coequalizer }q_a.}
$$
Equivalently, the necessary-and-sufficient conditions for a <left adjoint> with invertible unit are existence of the <coequalizers> in part (b) and their preservation by $G$. No preservation of arbitrary <coequalizers> is required.
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