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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?

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Recall the left triangle identity: $$\epsilon F \bullet F \eta = \textrm{id}_F$$ Since $\epsilon$ is pointwise monic, $\epsilon F$ is also pointwise monic. But $\epsilon F$ is split epic, so it must be an isomorphism with inverse $F \eta$. Since $F$ reflects isomorphisms, $\eta$ itself is an isomorphism. It immediately follows that $F$ is fully faithful. In particular, $\mathcal{C}$ is (equivalent to) a coreflective subcategory of $\mathcal{D}$.

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How do know counit is pointwise monic? I know there are some lightweight conditions that imply such equivalence (like pushouts in D), does something similar apply here? –  Marcin Łoś Feb 2 at 17:44
    
I think that's just imprecision in the statement of the hypotheses. –  Zhen Lin Feb 2 at 18:18

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