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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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    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".2012-01-16

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