= Solution
Assume $F$ is full and faithful and its image is closed under regular quotients. Its unit is invertible, hence monic. Factor the counit using the hypothesis on $\mathcal D$:
$$
FGB\overset{e}{\twoheadrightarrow}I\overset{m}{\hookrightarrow}B,
$$
with $e$ regular epic and $m$ monic. Closure gives $I\cong FA'$ for some $A'$. A right adjoint preserves <monomorphisms>: postcomposition with $Gm$ on $\mathcal C(X,-)$ corresponds under <adjunction> to postcomposition with $m$ on $\mathcal D(FX,-)$, which is injective. Thus $Gm$ is monic.
The triangle identity and invertibility of $\eta_{GB}$ give $G\varepsilon_B=(\eta_{GB})^{-1}$, so $Gm\,Ge$ is invertible. A <monomorphism> whose composite with another arrow is an <isomorphism> is itself split epic and hence an <isomorphism>; therefore both $Gm$ and $Ge$ are <isomorphisms>. The arrow $e$ is between objects in the essential image of the full and faithful $F$. After choosing $I\cong FA'$, write it as $Fh$. Naturality of the invertible unit identifies $GFh$ with $h$, so $Ge$ invertible implies $h$ and hence $e$ invertible. Now $\varepsilon_B=me$ is monic. This proves $\boxed{\text{(ii)}\Rightarrow\text{(i)}}$.
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