= Solution
Assume $L_2F=0$ for every additive <right exact functor> into an <abelian category>. Given a <subobject> $K\hookrightarrow P$ of a projective, let $X$ be its <cokernel in a category>. The sequence $0\to K\to P\to X\to0$ is exact, and dimension shifting gives $L_1F(K)\cong L_2F(X)=0$ for every $F$. The projectivity criterion proved in the root solution makes $K$ projective. This gives (ii)$\Rightarrow$(i).
For any short exact sequence $0\to X\to Y\to Z\to0$, the long exact sequence now starts at the relevant place with
$$
0\to L_1F(X)\to L_1F(Y)\to L_1F(Z).
$$
It is exact because the preceding term $L_2F(Z)$ vanishes. This is precisely left exactness of $L_1F$, proving $\boxed{\text{(ii)}\Rightarrow\text{(iii)}}$.
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