I wish to show that for an adjunction $F: \mathcal{C} \to \mathcal{D} \dashv G: \mathcal{D} \to \mathcal{C}$, if both the unit $\eta$ and the counit $\epsilon$ are monomorphisms and $\mathcal{C}$ is balanced (i.e. a monomorphic epimorphism is an isomorphism), then $F$ is full and faithful.
I have completed 2 preceding parts to this question: firstly, showing that if $F: \mathcal{C} \to \mathcal{D}$ is faithful and $\mathcal{C}$ balanced, then $F$ reflects isomorphisms. Secondly, showing that $F$ is faithful iff $\eta$ is a (pointwise) monomorphism. To show the latter, I had to look at the correspondence between $Ff$ and its corresponding map under the bijection (which is basically $\eta f$)
So, for this third part I guess I need to use the first part: we now know $F$ is faithful since $\eta$ monomorphic, and $\mathcal{C}$ is balanced so $F$ reflects isomorphisms: we now wish to show that for any $h: FA \to FB$, $h=Fg$ for some $g:A \to B$. I am not completely sure how we are going to use this reflectivity; perhaps we show that one of $\eta$ or $\epsilon$ is also an epimorphism and so invertible, then we can make use of this inverse by conjugating?
I played around with things for a bit but couldn't quite get anywhere. I have seen a similar result; $G$ is full and faithful if $\epsilon$ isomorphic, but in this case the initial assumption gives you an invertible morphism immediately (rather than 2 monomorphisms) so it seems easier. Could someone steer me in the right direction?