= Solution
Assume <subobjects> of projectives are projective. For any object $X$, choose $0\to K\to P\to X\to0$ with $P$ projective. The <kernel in a category> $K$ is projective by the assumption. The derived long exact sequence gives a dimension-shifting <isomorphism>
$$
L_2F(X)\cong L_1F(K),
$$
because $L_2F(P)=L_1F(P)=0$. The right side is zero because $K$ is projective. Since $X$ and the <right exact functor> $F$ were arbitrary, $\boxed{\text{(i)}\Rightarrow\text{(ii)}}$.
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