Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 26 / 4 / ii / 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 $\mathcal{D}$:with $e$ regular epic and $m$ monic. Closure gives $I\cong F{A}^{\prime }$ for some $A^{\prime }$. 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 $G{\epsilon }_B=({\eta }_{GB}{)}^{-1}$, so $GmGe$ 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 $I\cong F{A}^{\prime }$, write it as $Fh$. Naturality of the invertible unit identifies $GFh$ with $h$, so $Ge$ invertible implies $h$ and hence $e$ invertible. Now ${\epsilon }_B=me$ is monic. This proves $\boxed{\text{(ii)}\Rightarrow \text{(i)}}$.