 Solution

Assume subobjects of projectives are projective. For any object $X$, choose $0\to K\to P\to X\to 0$ with $P$ projective. The kernel in a category $K$ is projective by the assumption. The derived long exact sequence gives a dimension-shifting isomorphismbecause $L_{2}F(P)=L_{1}F(P)=0$. The right side is zero because $K$ is projective. Since $X$ and the right exact functor $F$ were arbitrary, $\boxed{\text{(i)}\Rightarrow \text{(ii)}}$.