= Solution
A <balanced category> is one in which every <morphism> that is both a <monomorphism> and an <epimorphism> is an <isomorphism>. A <faithful functor> reflects both of these cancellation properties: for instance, $fu=fv$ implies $Ff\,Fu=Ff\,Fv$, so monicity of $Ff$ and faithfulness imply $u=v$; the dual argument applies to <epimorphisms>. If $Ff$ is invertible, it is both monic and epic. Thus a <faithful functor> from a <balanced category> reflects <isomorphisms>.
For the <adjunction> $F\dashv G$, the <unit and counit of an adjunction> satisfy
$$
\varepsilon_{FA}F\eta_A=1_{FA},\qquad G\varepsilon_B\eta_{GB}=1_{GB}.
$$
If $F$ is a <faithful functor> and $\eta_Au=\eta_Av$, applying $F$ and composing with $\varepsilon_{FA}$ gives $Fu=Fv$, hence $u=v$. Thus each $\eta_A$ is a <monomorphism>. Conversely, if every $\eta_A$ is monic and $Fu=Fv$ for $u,v:X\to A$, <naturality> gives
$$
\eta_Au=GFu\,\eta_X=GFv\,\eta_X=\eta_Av,
$$
so $u=v$. This proves the <faithful left adjoint criterion>.
Now assume both $\eta$ and $\varepsilon$ are pointwise monic. The first triangle identity makes $\varepsilon_{FA}$ a <split epimorphism>; since it is also a <monomorphism>, it is invertible and $F\eta_A$ is its inverse. The <faithful left adjoint criterion> says that $F$ is faithful, and the <balanced category> argument above then reflects the invertibility of $F\eta_A$ to that of $\eta_A$.
For completeness, an invertible unit makes $F$ a <full and faithful functor>. For $h:FA\to FA'$, set $k=\eta_{A'}^{-1}G(h)\eta_A$. The triangle identity gives $F\eta_A=\varepsilon_{FA}^{-1}$, so <naturality> of $\varepsilon$ yields
$$
F(k)=\varepsilon_{FA'}FG(h)\varepsilon_{FA}^{-1}=h.
$$
Faithfulness was already proved.
Let $q:FA\to B$ be a <strong epimorphism>, and put
$$
t=FG(q)\varepsilon_{FA}^{-1}:FA\to FGB.
$$
Then <naturality> gives $\varepsilon_Bt=q$. The square with left edge $q$, right edge the <monomorphism> $\varepsilon_B$, top edge $t$, and bottom edge $1_B$ has a diagonal $s:B\to FGB$ by the defining lifting property of a <strong epimorphism>. Thus $\varepsilon_Bs=1_B$. A monic <split epimorphism> is invertible, so $B\cong FGB$. The essential image of $F$ is therefore closed under <strong quotients>.
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