Remark 21.36.3. Warning! Let $u : \mathcal{C} \to \mathcal{D}$, $g$, $\mathcal{O}_\mathcal {D}$, and $\mathcal{O}_\mathcal {C}$ be as in Lemma 21.36.2. In general it is not the case that the diagram

$\xymatrix{ D(\mathcal{O}_\mathcal {C}) \ar[r]_{Lg_!} \ar[d]_{forget} & D(\mathcal{O}_\mathcal {D}) \ar[d]^{forget} \\ D(\mathcal{C}) \ar[r]^{Lg^{Ab}_!} & D(\mathcal{D}) }$

commutes where the functor $Lg_!^{Ab}$ is the one constructed in Lemma 21.36.2 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 18.40.2), but this phenomenon can occur even if $g_! = g_!^{Ab}$! Namely, the construction of $Lg_!$ in the proof of Lemma 21.36.2 shows that $Lg_!$ agrees with $Lg_!^{\textit{Ab}}$ if and only if the canonical maps

$Lg^{Ab}_!j_{U!}\mathcal{O}_ U \longrightarrow j_{u(U)!}\mathcal{O}_{u(U)}$

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

$Lg_!^{Ab} \circ forget \longrightarrow forget \circ Lg_!$

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