Lemma 85.5.3. With notation and hypotheses as in Lemma 85.5.2. For $K \in D(\mathcal{C}_{total})$ we have $(g'_ n)^{-1}Rh_{total, *}K = Rh_{n, *}g_ n^{-1}K$.

**Proof.**
Let $\mathcal{I}^\bullet $ be a K-injective complex on $\mathcal{C}_{total}$ representing $K$. Then $g_ n^{-1}K$ is represented by $g_ n^{-1}\mathcal{I}^\bullet = \mathcal{I}_ n^\bullet $ which is K-injective by Lemma 85.3.6. We have $(g'_ n)^{-1}h_{total, *}\mathcal{I}^\bullet = h_{n, *}g_ n^{-1}\mathcal{I}_ n^\bullet $ by Lemma 85.5.2 which gives the desired equality.
$\square$

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