The Stacks project

Proof. Recall that $g_ n^* = g_ n^{-1}$ because $g_ n^{-1}\mathcal{O} = \mathcal{O}_ n$ by the construction in Lemma 85.6.1. In particular $g_ n^*$ is exact and $Lg_ n^*$ is given by applying $g_ n^*$ to any representative complex of modules. Similarly for $g'_ n$. There is a canonical base change map $(g'_ n)^*Rh_{total, *}K \to Rh_{n, *}g_ n^*K$, see Cohomology on Sites, Remark 21.19.3. By Cohomology on Sites, Lemma 21.20.7 the image of this in $D(\mathcal{C}'_ n)$ is the map $(g'_ n)^{-1}Rh_{total, *}K_{ab} \to Rh_{n, *}g_ n^{-1}K_{ab}$ where $K_{ab}$ is the image of $K$ in $D(\mathcal{C}_{total})$. This we proved to be an isomorphism in Lemma 85.5.3 and the result follows. $\square$