I know that if a left adjoint is exact then the right adjoint preserves injectives.
For the converse, I read in "A QUICK PROOF OF THE GABRIEL-POPESCU THEOREM" of Mitchell that:
"a left adjoint whose codomain has enough injectives preserves monomorphisms if and only if its right adjoint preserves injectives".
So, in this case, the converse is true but I don't find any proof of this. Can someone help me?
Thank you so much!
The calculations (switching back and forth via the adjunction) are similar for both directions, so I will only provide a hint, since I guess you can take it from there.
Let $F \colon {\cal X} \to {\cal A} $ be left adjoint to $G \colon {\cal A} \to {\cal X} $, and let $ u \colon X \to Y $ be a monomorphism in ${\cal X}$.
(1) by assumption, we can embed $FX$ in some injective object $E$, i.e. there is a monomorphism $h \colon FX \to E$. Let $ \hat{h} \colon X \to GE $ be its adjoint.
(2) because $GE$ is injective, we can extend $\hat{h}$ along the monomorphism $u$.
(3) switch back via the adjunction to obtain a morphism $f \colon FY \to E$ with $h = f \circ Fu$ and conclude that $Fu$ must be monic.