The Stacks project

Proof. We have $g^* = Lg^* = g^{-1}$ and $(g')^* = L(g')^* = (g')^{-1}$ by condition (5). By Lemma 21.20.7 it suffices to prove the result on the derived category $D^+(\mathcal{C})$ or $D(\mathcal{C})$ of abelian sheaves.

Choose an object $K \in D^+(\mathcal{C})$. Let $\mathcal{I}^\bullet$ be a bounded below complex of injective abelian sheaves on $\mathcal{C}$ representing $K$. By Lemma 21.37.1 we see that $H^ p(U', (g')^{-1}\mathcal{I}^ q) = 0$ for all $p > 0$ and any $q$ and any $U' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$. Recall that $R^ pf'_*(g')^{-1}\mathcal{I}^ q$ is the sheaf associated to the presheaf $V' \mapsto H^ p(u'(V'), (g')^{-1}\mathcal{I}^ q)$, see Lemma 21.7.4. Thus we see that $(g')^{-1}\mathcal{I}^ q$ is right acyclic for the functor $f'_*$. By Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) we find that $f'_*(g')^*\mathcal{I}^\bullet$ represents $Rf'_*(g')^{-1}K$. By Modules on Sites, Lemma 18.41.4 we find that $f'_*(g')^{-1}\mathcal{I}^\bullet = g^{-1}f_*\mathcal{I}^\bullet$. Since $g^{-1}f_*\mathcal{I}^\bullet$ represents $g^{-1}Rf_*K$ we conclude.

Choose an object $K \in D(\mathcal{C})$. Let $\mathcal{I}^\bullet$ be a K-injective complex of abelian sheaves on $\mathcal{C}$ representing $K$. By Derived Categories, Lemma 13.31.9 and assumption (6) we find that $(g')^{-1}\mathcal{I}^\bullet$ is a K-injective complex of abelian sheaves on $\mathcal{C}'$. By Modules on Sites, Lemma 18.41.4 we find that $f'_*(g')^{-1}\mathcal{I}^\bullet = g^{-1}f_*\mathcal{I}^\bullet$. Since $f_*\mathcal{I}^\bullet$ represents $Rf_*K$ and since $f'_*(g')^{-1}\mathcal{I}^\bullet$ represents $Rf'_*(g')^{-1}K$ we conclude. $\square$