Positive right derived functors vanish on injective objects

The coaugmented complex $0\to J\stackrel{1_J}{\to }J\to 0\to 0\to \cdots$ is exact and all its terms are injective by [L2], so [L1] makes it an injective resolution of $J$. After applying $F$, its deleted cochain complex has only one nonzero term, namely $F(J)$ in degree $0$.

Let $I^{\mathrm{\prime }}$ be the supplied injective resolution datum on the same domain as $I$ that agrees with $I$ away from $J$ and assigns the trivial injective resolution from step 1.1 to $J$. By [L3], the right derived object computed from $I$ is isomorphic to the one computed from $I^{\mathrm{\prime }}$. By [L4], the complex computing $R_{I^{\mathrm{\prime }}}^{n}F(J)$ is the one-term complex from step 1.1, whose cohomology is zero in every positive degree. Hence $R_I^{n}F(J)=0$ for $n>0$.