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Let $A$ and $B$ be two abelian categories. Assume that there exist a functor $F$ between them which is exact, full and essencially surjective.

If $x$ is a projective object in $A$, then $F(x)$ is a projective object in $B$?

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I think you might need some faithfulness hypotheses for this to be true: the main difficulty is in lifting epimorphisms in $B$ to epimorphisms in $A$. –  Zhen Lin Jan 15 '12 at 9:06
But the property that $F$ is essentially surjective doesn't help to do this? –  Iihon Mckort Jan 15 '12 at 19:42
No, being essentially surjective is a condition on objects, not arrows. It's not obvious to me whether your combination of hypotheses implies that $F$ reflects epimorphisms. –  Zhen Lin Jan 15 '12 at 19:48
Do you know some hypothesis which reflects epimorphisms? –  Iihon Mckort Jan 16 '12 at 3:39
Well, adding "faithful" to your hypotheses will do it, but then $F$ becomes an equivalence of categories! Another possibility is to strengthen your exactness hypothesis from "exact" to "faithfully exact". –  Zhen Lin Jan 16 '12 at 10:24

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