The Stacks project

Remark 21.37.3. Warning! Let $u : \mathcal{C} \to \mathcal{D}$, $g$, $\mathcal{O}_\mathcal {D}$, and $\mathcal{O}_\mathcal {C}$ be as in Lemma 21.37.2. In general it is not the case that the diagram

commutes where the functor $Lg_!^{Ab}$ is the one constructed in Lemma 21.37.2 but using the constant sheaf $\mathbf{Z}$ as the structure sheaf on both $\mathcal{C}$ and $\mathcal{D}$. In general it isn't even the case that $g_! = g_!^{Ab}$ (see Modules on Sites, Remark 18.41.2), but this phenomenon can occur even if $g_! = g_!^{Ab}$! Namely, the construction of $Lg_!$ in the proof of Lemma 21.37.2 shows that $Lg_!$ agrees with $Lg_!^{\textit{Ab}}$ if and only if the canonical maps

are isomorphisms in $D(\mathcal{D})$ for all objects $U$ in $\mathcal{C}$. In general all we can say is that there exists a natural transformation