hcju/stacksqa huggingface dataset:
By our construction of derived pullback in Lemma \ref{lemma-derived-base-change}. and the existence of resolutions in Lemma \ref{lemma-K-flat-resolution} we may replace $\mathcal{F}^\bullet$ and $\mathcal{G}^\bullet$ by complexes of $\mathcal{O}'$-modules which are K-flat and have flat terms. In this case $\mathcal{F}^\bullet \otimes_{\mathcal{O}'}^{\mathbf{L}} \mathcal{G}^\bullet$ is just the total complex associated to the double complex $\mathcal{F}^\bullet \otimes_{\mathcal{O}'} \mathcal{G}^\bullet$. The complex $\text{Tot}(\mathcal{F}^\bullet \otimes_{\mathcal{O}'} \mathcal{G}^\bullet)$ is K-flat with flat terms by Lemma \ref{lemma-tensor-product-K-flat} and Modules on Sites, Lemma \ref{sites-modules-lemma-tensor-flats}. Hence the isomorphism of the lemma comes from the isomorphism $$ \text{Tot}(f^*\mathcal{F}^\bullet \otimes_{\mathcal{O}} f^*\mathcal{G}^\bullet) \longrightarrow f^*\text{Tot}(\mathcal{F}^\bullet \otimes_{\mathcal{O}'} \mathcal{G}^\bullet) $$ whose constituents are the isomorphisms $f^*\mathcal{F}^p \otimes_{\mathcal{O}} f^*\mathcal{G}^q \to f^*(\mathcal{F}^p \otimes_{\mathcal{O}'} \mathcal{G}^q)$ of Modules on Sites, Lemma \ref{sites-modules-lemma-tensor-product-pullback}.
Proof of (2). If $\mathcal{C}$ has enough points and $\mathcal{K}_p^\bullet$ is K-flat for all points $p$ of $\mathcal{C}$ then we see that $\mathcal{K}^\bullet$ is K-flat because $\otimes$ and direct sums commute with taking stalks and because we can check exactness at stalks, see Modules on Sites, Lemma \ref{sites-modules-lemma-check-exactness-stalks}. \medskip\noindent Proof of (1). Assume $\mathcal{K}^\bullet$ is K-flat. Choose a quasi-isomorphism $a : \mathcal{L}^\bullet \to \mathcal{K}^\bullet$ such that $\mathcal{L}^\bullet$ is K-flat with flat terms, see Lemma \ref{lemma-K-flat-resolution}. Any pullback of $\mathcal{L}^\bullet$ is K-flat, see Lemma \ref{lemma-pullback-K-flat}. In particular the stalk $\mathcal{L}_p^\bullet$ is a K-flat complex of $\mathcal{O}_p$-modules. Thus the cone $C(a)$ on $a$ is a K-flat (Lemma \ref{lemma-K-flat-two-out-of-three}) acyclic complex of $\mathcal{O}$-modules and it suffuces to show the stalk of $C(a)$ is K-flat (by More on Algebra, Lemma \ref{more-algebra-lemma-K-flat-two-out-of-three}). Thus we may assume that $\mathcal{K}^\bullet$ is K-flat and acyclic. \medskip\noindent Assume $\mathcal{K}^\bullet$ is acyclic and K-flat. Before continuing we replace the site $\mathcal{C}$ by another one as in Sites, Lemma \ref{sites-lemma-topos-good-site} to insure that $\mathcal{C}$ has all finite limits. This implies the category of neighbourhoods of $p$ is filtered (Sites, Lemma \ref{sites-lemma-neighbourhoods-directed}) and the colimit defining the stalk of a sheaf is filtered. Let $M$ be a finitely presented $\mathcal{O}_p$-module. It suffices to show that $\mathcal{K}^\bullet \otimes_{\mathcal{O}_p} M$ is acyclic, see More on Algebra, Lemma \ref{more-algebra-lemma-universally-acyclic-K-flat}. Since $\mathcal{O}_p$ is the filtered colimit of $\mathcal{O}(U)$ where $U$ runs over the neighbourhoods of $p$, we can find a neighbourhood $(U, x)$ of $p$ and a finitely presented $\mathcal{O}(U)$-module $M'$ whose base change to $\mathcal{O}_p$ is $M$, see Algebra, Lemma \ref{algebra-lemma-colimit-category-fp-modules}. By Lemma \ref{lemma-restriction-K-flat} we may replace $\mathcal{C}, \mathcal{O}, \mathcal{K}^\bullet$ by $\mathcal{C}/U, \mathcal{O}_U, \mathcal{K}^\bullet|_U$. We conclude that we may assume there exists an $\mathcal{O}$-module $\mathcal{F}$ such that $M \cong \mathcal{F}_p$. Since $\mathcal{K}^\bullet$ is K-flat and acyclic, we see that $\mathcal{K}^\bullet \otimes_\mathcal{O} \mathcal{F}$ is acyclic (as it computes the derived tensor product by definition). Taking stalks is an exact functor, hence we get that $\mathcal{K}^\bullet \otimes_{\mathcal{O}_p} M$ is acyclic as desired.
We will use the existence of K-flat resolutions with flat terms (Lemma \ref{lemma-K-flat-resolution}), we will use that derived pullback is computed by such complexes (Lemma \ref{lemma-derived-base-change}), and that pullbacks preserve these properties (Lemma \ref{lemma-pullback-K-flat}). If we choose such resolutions $\mathcal{P}^\bullet \to \mathcal{K}^\bullet$ and $\mathcal{Q}^\bullet \to \mathcal{M}^\bullet$, then we see that $$ \xymatrix{ Lf^*\text{Tot}(\mathcal{P}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} \mathcal{Q}^\bullet) \ar[r] \ar[d] & Lf^*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} \mathcal{M}^\bullet) \ar[d] \\ f^*\text{Tot}(\mathcal{P}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} \mathcal{Q}^\bullet) \ar[d] \ar[r] & f^*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{D}} \mathcal{M}^\bullet) \ar[d] \\ \text{Tot}(f^*\mathcal{P}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} f^*\mathcal{Q}^\bullet) \ar[r] & \text{Tot}(f^*\mathcal{K}^\bullet \otimes_{\mathcal{O}_\mathcal{C}} f^*\mathcal{M}^\bullet) } $$ commutes. However, now the left hand side of the diagram is the left hand side of the diagram by our choice of $\mathcal{P}^\bullet$ and $\mathcal{Q}^\bullet$ and Lemma \ref{lemma-tensor-product-K-flat}.
We will use the existence of K-flat resolutions and K-injective resolutions, see Lemmas \ref{lemma-K-flat-resolution}, \ref{lemma-derived-base-change}, and \ref{lemma-pullback-K-flat} and the discussion above. Choose a quasi-isomorphism $\mathcal{K}^\bullet \to \mathcal{I}^\bullet$ where $\mathcal{I}^\bullet$ is K-injective as a complex of $\mathcal{O}_\mathcal{C}$-modules. Choose a quasi-isomorphism $\mathcal{Q}^\bullet \to f_*\mathcal{I}^\bullet$ where $\mathcal{Q}^\bullet$ is a K-flat complex of $\mathcal{O}_\mathcal{D}$-modules with flat terms. We can choose a K-flat complex of $\mathcal{O}_\mathcal{D}$-modules $\mathcal{P}^\bullet$ with flat terms and a diagram of morphisms of complexes $$ \xymatrix{ \mathcal{P}^\bullet \ar[r] \ar[d] & f_*\mathcal{K}^\bullet \ar[d] \\ \mathcal{Q}^\bullet \ar[r] & f_*\mathcal{I}^\bullet } $$ commutative up to homotopy where the top horizontal arrow is a quasi-isomorphism. Namely, we can first choose such a diagram for some complex $\mathcal{P}^\bullet$ because the quasi-isomorphisms form a multiplicative system in the homotopy category of complexes and then we can choose a resolution of $\mathcal{P}^\bullet$ by a K-flat complex with flat terms. Taking pullbacks we obtain a diagram of morphisms of complexes $$ \xymatrix{ f^*\mathcal{P}^\bullet \ar[r] \ar[d] & f^*f_*\mathcal{K}^\bullet \ar[d] \ar[r] & \mathcal{K}^\bullet \ar[d] \\ f^*\mathcal{Q}^\bullet \ar[r] & f^*f_*\mathcal{I}^\bullet \ar[r] & \mathcal{I}^\bullet } $$ commutative up to homotopy. The outer rectangle witnesses the truth of the statement in the lemma.
A key observation is that an injective abelian sheaf $\mathcal{I}$ is divisible. Namely, if $s \in \mathcal{I}(U)$ is a local section, then we interpret $s$ as a map $s : j_{U!}\mathbf{Z} \to \mathcal{I}$ and we apply the defining property of an injective object to the injective map of sheaves $n : j_{U!}\mathbf{Z} \to j_{U!}\mathbf{Z}$ to see that there exists an $s' \in \mathcal{I}(U)$ with $ns' = s$. \medskip\noindent For a sheaf $\mathcal{F}$ denote $\mathcal{F}_{tor}$ its torsion subsheaf. We claim that if $\mathcal{I}^\bullet$ is a complex of injective abelian sheaves whose cohomology sheaves are torsion, then $$ \mathcal{I}^\bullet_{tor} \to \mathcal{I}^\bullet $$ is a quasi-isomorphism. Namely, by flatness of $\mathbf{Q}$ over $\mathbf{Z}$ we have $$ H^p(\mathcal{I}^\bullet) \otimes_\mathbf{Z} \mathbf{Q} = H^p(\mathcal{I}^\bullet \otimes_\mathbf{Z} \mathbf{Q}) $$ which is zero because the cohomology sheaves are torsion. By divisibility (shown above) we see that $\mathcal{I}^\bullet \to \mathcal{I}^\bullet \otimes_\mathbf{Z} \mathbf{Q}$ is surjective with kernel $\mathcal{I}^\bullet_{tor}$. The claim follows from the long exact sequence of cohomology sheaves associated to the short exact sequence you get. \medskip\noindent To prove the lemma we will construct right adjoint $T : D(\mathcal{C}) \to D(\mathcal{A})$. Namely, given $K$ in $D(\mathcal{C})$ we can represent $K$ by a K-injective complex $\mathcal{I}^\bullet$ whose cohomology sheaves are injective, see Injectives, Theorem \ref{injectives-theorem-K-injective-embedding-grothendieck}. Then we set $T(K) = \mathcal{I}^\bullet_{tor}$, in other words, $T$ is the right derived functor of taking torsion. The functor $T$ is a right adjoint to $i : D(\mathcal{A}) \to D_\mathcal{A}(\mathcal{C})$. This readily follows from the observation that if $\mathcal{F}^\bullet$ is a complex of torsion sheaves, then $$ \Hom_{K(\mathcal{A})}(\mathcal{F}^\bullet, I^\bullet_{tor}) = \Hom_{K(\textit{Ab}(\mathcal{C}))}(\mathcal{F}^\bullet, I^\bullet) $$ in particular $\mathcal{I}^\bullet_{tor}$ is a K-injective complex of $\mathcal{A}$. Some details omitted; in case of doubt, it also follows from the more general Derived Categories, Lemma \ref{derived-lemma-derived-adjoint-functors}. Our claim above gives that $L = T(i(L))$ for $L$ in $D(\mathcal{A})$ and $i(T(K)) = K$ if $K$ is in $D_\mathcal{A}(\mathcal{C})$. Using Categories, Lemma \ref{categories-lemma-adjoint-fully-faithful} the result follows.
Let $\mathcal{I}^\bullet$ be a K-injective complex of $\mathcal{O}$-modules representing $K$. Then $$ H^q(U, K) = H^q(\Gamma(U, \mathcal{I}^\bullet)) = H^q(\Gamma(\mathcal{C}/U, \mathcal{I}^\bullet|_{\mathcal{C}/U})) $$ by construction of cohomology. By Lemma \ref{lemma-restrict-K-injective-to-open} the complex $\mathcal{I}^\bullet|_{\mathcal{C}/U}$ is a K-injective complex representing $K|_{\mathcal{C}/U}$ and the lemma follows.
The equality $H^q(U, K) = H^q(\mathcal{C}/U, K|_{\mathcal{C}/U})$ holds by Lemma \ref{lemma-unbounded-cohomology-of-open}. Choose a K-injective complex $\mathcal{I}^\bullet$ representing $K$. Then $$ H^q(U, K) = \frac{\Ker(\mathcal{I}^q(U) \to \mathcal{I}^{q + 1}(U))} {\Im(\mathcal{I}^{q - 1}(U) \to \mathcal{I}^q(U))}. $$ by our construction of cohomology. Since $H^q(K) = \Ker(\mathcal{I}^q \to \mathcal{I}^{q + 1})/ \Im(\mathcal{I}^{q - 1} \to \mathcal{I}^q)$ the result is clear.
Represent $E$ by a K-injective complex $\mathcal{I}^\bullet$ of $\mathcal{O}_\mathcal{C}$-modules. Then $Rf_*(E) = f_*\mathcal{I}^\bullet$ and $Rg_*(E|_{\mathcal{C}/U}) = g_*(\mathcal{I}^\bullet|_{\mathcal{C}/U})$ by Lemma \ref{lemma-restrict-K-injective-to-open}. Since it is clear that $(f_*\mathcal{F})|_{\mathcal{D}/V} = g_*(\mathcal{F}|_{\mathcal{C}/U})$ for any sheaf $\mathcal{F}$ on $\mathcal{C}$ (see Modules on Sites, Lemma \ref{sites-modules-lemma-localize-morphism-ringed-sites} or the more basic Sites, Lemma \ref{sites-lemma-localize-morphism}) the result follows.
Consider the punctual topos $pt$ endowed with $\mathcal{O}_{pt}$ given by the ring $\Gamma(\mathcal{O}_\mathcal{D})$. There is a canonical morphism $(\mathcal{D}, \mathcal{O}_\mathcal{D}) \to (pt, \mathcal{O}_{pt})$ of ringed topoi inducing the identification on global sections of structure sheaves. Then $D(\mathcal{O}_{pt}) = D(\Gamma(\mathcal{O}_\mathcal{D}))$. The assertion $R\Gamma(\mathcal{D}, -) \circ Rf_* = R\Gamma(\mathcal{C}, -)$ follows from Lemma \ref{lemma-derived-pushforward-composition} applied to $$ (\mathcal{C}, \mathcal{O}_\mathcal{C}) \to (\mathcal{D}, \mathcal{O}_\mathcal{D}) \to (pt, \mathcal{O}_{pt}) $$ The second (more general) statement follows from the first statement after applying Lemma \ref{lemma-restrict-direct-image-open}.
The map is constructed as follows. Choose a K-injective complex $\mathcal{I}^\bullet$ representing $K$. Choose a quasi-isomorpism $\mathcal{I}^\bullet \to \mathcal{J}^\bullet$ where $\mathcal{J}^\bullet$ is a K-injective complex of abelian groups. Then the map in (1) is given by $\Gamma(\mathcal{C}, \mathcal{I}^\bullet) \to \Gamma(\mathcal{C}, \mathcal{J}^\bullet)$ (2) is given by $\Gamma(U, \mathcal{I}^\bullet) \to \Gamma(U, \mathcal{J}^\bullet)$ and the map in (3) is given by $f_*\mathcal{I}^\bullet \to f_*\mathcal{J}^\bullet$. To show that these maps are isomorphisms, it suffices to prove they induce isomorphisms on cohomology groups and cohomology sheaves. By Lemmas \ref{lemma-unbounded-cohomology-of-open} and \ref{lemma-unbounded-describe-higher-direct-images} it suffices to show that the map $$ H^0(\mathcal{C}, K) \longrightarrow H^0(\mathcal{C}, K_{ab}) $$ is an isomorphism. Observe that $$ H^0(\mathcal{C}, K) = \Hom_{D(\mathcal{O}_\mathcal{C})}(\mathcal{O}_\mathcal{C}, K) $$ and similarly for the other group. Choose any complex $\mathcal{K}^\bullet$ of $\mathcal{O}_\mathcal{C}$-modules representing $K$. By construction of the derived category as a localization we have $$ \Hom_{D(\mathcal{O}_\mathcal{C})}(\mathcal{O}_\mathcal{C}, K) = \colim_{s : \mathcal{F}^\bullet \to \mathcal{O}_\mathcal{C}} \Hom_{K(\mathcal{O}_\mathcal{C})}(\mathcal{F}^\bullet, \mathcal{K}^\bullet) $$ where the colimit is over quasi-isomorphisms $s$ of complexes of $\mathcal{O}_\mathcal{C}$-modules. Similarly, we have $$ \Hom_{D(\underline{\mathbf{Z}}_\mathcal{C})} (\underline{\mathbf{Z}}_\mathcal{C}, K) = \colim_{s : \mathcal{G}^\bullet \to \underline{\mathbf{Z}}_\mathcal{C}} \Hom_{K(\underline{\mathbf{Z}}_\mathcal{C})} (\mathcal{G}^\bullet, \mathcal{K}^\bullet) $$ Next, we observe that the quasi-isomorphisms $s : \mathcal{G}^\bullet \to \underline{\mathbf{Z}}_\mathcal{C}$ with $\mathcal{G}^\bullet$ bounded above complex of flat $\underline{\mathbf{Z}}_\mathcal{C}$-modules is cofinal in the system. (This follows from Modules on Sites, Lemma \ref{sites-modules-lemma-module-quotient-flat} and Derived Categories, Lemma \ref{derived-lemma-subcategory-left-resolution}; see discussion in Section \ref{section-flat}.) Hence we can construct an inverse to the map $H^0(\mathcal{C}, K) \longrightarrow H^0(\mathcal{C}, K_{ab})$ by representing an element $\xi \in H^0(\mathcal{C}, K_{ab})$ by a pair $$ (s : \mathcal{G}^\bullet \to \underline{\mathbf{Z}}_\mathcal{C}, a : \mathcal{G}^\bullet \to \mathcal{K}^\bullet) $$ with $\mathcal{G}^\bullet$ a bounded above complex of flat $\underline{\mathbf{Z}}_\mathcal{C}$-modules and sending this to $$ (\mathcal{G}^\bullet \otimes_{\underline{\mathbf{Z}}_\mathcal{C}} \mathcal{O}_\mathcal{C} \to \mathcal{O}_\mathcal{C}, \mathcal{G}^\bullet  \otimes_{\underline{\mathbf{Z}}_\mathcal{C}} \mathcal{O}_\mathcal{C} \to \mathcal{K}^\bullet) $$ The only thing to note here is that the first arrow is a quasi-isomorphism by Lemmas \ref{lemma-derived-tor-quasi-isomorphism-other-side} and \ref{lemma-bounded-flat-K-flat}. We omit the detailed verification that this construction is indeed an inverse.
We have to show that $$ \Hom_{D(\mathcal{O})}(j_!E, F) = \Hom_{D(\mathcal{O}_U)}(E, F|_U) $$ Choose a complex $\mathcal{E}^\bullet$ of $\mathcal{O}_U$-modules representing $E$ and choose a K-injective complex $\mathcal{I}^\bullet$ representing $F$. By Lemma \ref{lemma-restrict-K-injective-to-open} the complex $\mathcal{I}^\bullet|_U$ is K-injective as well. Hence we see that the formula above becomes $$ \Hom_{D(\mathcal{O})}(j_!\mathcal{E}^\bullet, \mathcal{I}^\bullet) = \Hom_{D(\mathcal{O}_U)}(\mathcal{E}^\bullet, \mathcal{I}^\bullet|_U) $$ which holds as $|_U$ and $j_!$ are adjoint functors (Modules on Sites, Lemma \ref{sites-modules-lemma-extension-by-zero}) and Derived Categories, Lemma \ref{derived-lemma-K-injective}.
This is true because $$ \Hom_{K(\mathcal{O}_\mathcal{D})}(\mathcal{F}^\bullet, f_*\mathcal{I}^\bullet) = \Hom_{K(\mathcal{O}_\mathcal{C})}(f^*\mathcal{F}^\bullet, \mathcal{I}^\bullet) $$ by Modules on Sites, Lemma \ref{sites-modules-lemma-adjoint-pullback-pushforward-modules} and the fact that $f^*$ is exact as $f$ is assumed to be flat.
This is true because $\Hom_{K(\mathcal{O}')}(\mathcal{G}^\bullet, \Hom_\mathcal{O}(\mathcal{O}', \mathcal{I}^\bullet)) = \Hom_{K(\mathcal{O})}(\mathcal{G}^\bullet, \mathcal{I}^\bullet)$ by Modules on Sites, Lemma \ref{sites-modules-lemma-adjoint-hom-restrict}.
Let $E \in D(\mathcal{C}/X)$. Choose a K-injective complex $\mathcal{I}^\bullet$ of abelian sheaves on $\mathcal{C}/X$ representing $E$. By Lemma \ref{lemma-restrict-K-injective-to-open} we see that $j_{X'/X}^{-1}\mathcal{I}^\bullet$ is K-injective too. Hence we may compute $Rj_{X'/Y'}(j_{X'/X}^{-1}E)$ by $j_{X'/Y', *}j_{X'/X}^{-1}\mathcal{I}^\bullet$. Thus we see that the equality holds by Sites, Lemma \ref{sites-lemma-localize-cartesian-square}.
Since $j_{Y'/Y}^{-1}\mathcal{O}_Y = \mathcal{O}_{Y'}$ we have $j_{Y'/Y}^* = Lj_{Y'/Y}^* = j_{Y'/Y}^{-1}$. Similarly we have $j_{X'/X}^* = Lj_{X'/X}^* = j_{X'/X}^{-1}$. Thus by Lemma \ref{lemma-modules-abelian-unbounded} it suffices to prove the result on derived categories of abelian sheaves which we did in Lemma \ref{lemma-localize-cartesian-square}.
To calculate $R\lim$ on an object $K$ of $D(\mathcal{C} \times \mathbf{N})$ we choose a K-injective representative $\mathcal{I}^\bullet$ whose terms are injective objects of $\textit{Ab}(\mathcal{C} \times \mathbf{N})$, see Injectives, Theorem \ref{injectives-theorem-K-injective-embedding-grothendieck}. We may and do think of $\mathcal{I}^\bullet$ as an inverse system of complexes $(\mathcal{I}_n^\bullet)$ and then we see that $$ R\lim K = \lim \mathcal{I}_n^\bullet $$ where the right hand side is the termwise inverse limit. \medskip\noindent Let $\mathcal{J} = (\mathcal{J}_n)$ be an injective object of $\textit{Ab}(\mathcal{C} \times \mathbf{N})$. The morphisms $(U, n) \to (U, n + 1)$ are monomorphisms of $\mathcal{C} \times \mathbf{N}$, hence $\mathcal{J}(U, n + 1) \to \mathcal{J}(U, n)$ is surjective (Lemma \ref{lemma-restriction-along-monomorphism-surjective}). It follows that $\mathcal{J}_{n + 1} \to \mathcal{J}_n$ is surjective as a map of presheaves. \medskip\noindent Note that the functor $i_n^{-1}$ has an exact left adjoint $i_{n, !}$. Namely, $i_{n, !}\mathcal{F}$ is the inverse system $\ldots 0 \to 0 \to \mathcal{F} \to \ldots \to \mathcal{F}$. Thus the complexes $i_n^{-1}\mathcal{I}^\bullet = \mathcal{I}_n^\bullet$ are K-injective by Derived Categories, Lemma \ref{derived-lemma-adjoint-preserve-K-injectives}. \medskip\noindent Because we chose our K-injective complex to have injective terms we conclude that $$ 0 \to  \lim \mathcal{I}_n^\bullet \to \prod \mathcal{I}_n^\bullet \to \prod \mathcal{I}_n^\bullet \to 0 $$ is a short exact sequence of complexes of abelian sheaves as it is a short exact sequence of complexes of abelian presheaves. Moreover, the products in the middle and the right represent the products in $D(\mathcal{C})$, see Injectives, Lemma \ref{injectives-lemma-derived-products} and its proof (this is where we use that $\mathcal{I}_n^\bullet$ is K-injective). Thus $R\lim K$ is a homotopy limit of the inverse system $(K_n)$ by definition of homotopy limits in triangulated categories.
Let $(K_n)$ be an inverse system of objects of $D(\mathcal{O})$. By induction on $n$ we may choose actual complexes $\mathcal{K}_n^\bullet$ of $\mathcal{O}$-modules and maps of complexes $\mathcal{K}_{n + 1}^\bullet \to \mathcal{K}_n^\bullet$ representing the maps $K_{n + 1} \to K_n$ in $D(\mathcal{O})$. In other words, there exists an object $K$ in $D(\mathcal{C} \times \mathbf{N})$ whose associated inverse system is the given one. Next, consider the commutative diagram $$ \xymatrix{ \Sh(\mathcal{C} \times \mathbf{N}) \ar[r]_g \ar[d]_{f \times 1} & \Sh(\mathcal{C}) \ar[d]_f \\ \Sh(\mathcal{C}' \times \mathbf{N}) \ar[r]^{g'} & \Sh(\mathcal{C}') } $$ of morphisms of topoi. It follows that $R\lim R(f \times 1)_*K = Rf_* R\lim K$. Working through the definitions and using Lemma \ref{lemma-derived-limit-is-ok} we obtain that $R\lim (Rf_*K_n) = Rf_*(R\lim K_n)$. \medskip\noindent Alternate proof in case $\mathcal{C}$ has enough points. Consider the defining distinguished triangle $$ R\lim K_n \to \prod K_n \to \prod K_n $$ in $D(\mathcal{O})$. Applying the exact functor $Rf_*$ we obtain the distinguished triangle $$ Rf_*(R\lim K_n) \to Rf_*\left(\prod K_n\right) \to Rf_*\left(\prod K_n\right) $$ in $D(\mathcal{O}')$. Thus we see that it suffices to prove that $Rf_*$ commutes with products in the derived category (which are not just given by products of complexes, see Injectives, Lemma \ref{injectives-lemma-derived-products}). However, since $Rf_*$ is a right adjoint by Lemma \ref{lemma-adjoint} this follows formally (see Categories, Lemma \ref{categories-lemma-adjoint-exact}). Caution: Note that we cannot apply Categories, Lemma \ref{categories-lemma-adjoint-exact} directly as $R\lim K_n$ is not a limit in $D(\mathcal{O})$.
Let $\gamma \in H^m(R\lim K_n)(V)$ map to zero in $H^m(K_{n(V)})(V)$. Since $H^m(R\lim K_n)$ is the sheafification of $U \mapsto H^m(U, R\lim K_n)$ (by Lemma \ref{lemma-sheafification-cohomology}) we can choose $\{V_i \to V\} \in \text{Cov}_V$ and elements $\tilde\gamma_i \in H^m(V_i, R\lim K_n)$ mapping to $\gamma|_{V_i}$. Then $\tilde\gamma_i$ maps to $\tilde\gamma_{i, n(V)} \in H^m(V_i, K_{n(V)})$. Using that $H^m(K_{n(V)})$ is the sheafification of $U \mapsto H^m(U, K_{n(V)})$ (by Lemma \ref{lemma-sheafification-cohomology} again) we see that after replacing $\{V_i \to V\}$ by a refinement we may assume that $\tilde\gamma_{i, n(V)} = 0$ for all $i$. For this covering we consider the short exact sequences $$ 0 \to R^1\lim H^{m - 1}(V_i, K_n) \to H^m(V_i, R\lim K_n) \to \lim H^m(V_i, K_n) \to 0 $$ of Lemma \ref{lemma-RGamma-commutes-with-Rlim}. By assumption (1) the group on the left is zero and by assumption (2) the group on the right maps injectively into $H^m(V_i, K_{n(V)})$. We conclude $\tilde\gamma_i = 0$ and hence $\gamma = 0$ as desired.
Set $K_n = \tau_{\geq -n}E$ and $K = R\lim K_n$. The canonical map $E \to K$ comes from the canonical maps $E \to K_n = \tau_{\geq -n}E$. We have to show that $E \to K$ induces an isomorphism $H^m(E) \to H^m(K)$ of cohomology sheaves. In the rest of the proof we fix $m$. If $n \geq -m$, then the map $E \to \tau_{\geq -n}E = K_n$ induces an isomorphism $H^m(E) \to H^m(K_n)$. To finish the proof it suffices to show that for every $V \in \mathcal{B}$ there exists an integer $n(V) \geq -m$ such that the map $H^m(K)(V) \to H^m(K_{n(V)})(V)$ is injective. Namely, then the composition $$ H^m(E)(V) \to H^m(K)(V) \to H^m(K_{n(V)})(V) $$ is a bijection and the second arrow is injective, hence the first arrow is bijective. By property (1) this will imply $H^m(E) \to H^m(K)$ is an isomorphism. Set $$ n(V) = 1 + \max\{-m, p(V, m - 1) - m, -1 + p(V, m) - m, -2 + p(V, m + 1) - m\}. $$ so that in any case $n(V) \geq -m$. Claim: the maps $$ H^{m - 1}(V_i, K_{n + 1}) \to H^{m - 1}(V_i, K_n) \quad\text{and}\quad H^m(V_i, K_{n + 1}) \to H^m(V_i, K_n) $$ are isomorphisms for $n \geq n(V)$ and $\{V_i \to V\} \in \text{Cov}_V$. The claim implies conditions (1) and (2) of Lemma \ref{lemma-cohomology-derived-limit-injective} are satisfied and hence implies the desired injectivity. Recall (Derived Categories, Remark \ref{derived-remark-truncation-distinguished-triangle}) that we have distinguished triangles $$ H^{-n - 1}(E)[n + 1] \to K_{n + 1} \to K_n \to H^{-n - 1}(E)[n + 2] $$ Looking at the asssociated long exact cohomology sequence the claim follows if $$ H^{m + n}(V_i, H^{-n - 1}(E)),\quad H^{m + n + 1}(V_i, H^{-n - 1}(E)),\quad H^{m + n + 2}(V_i, H^{-n - 1}(E)) $$ are zero for $n \geq n(V)$ and $\{V_i \to V\} \in \text{Cov}_V$. This follows from our choice of $n(V)$ and the assumption in the lemma.
Observe that $K = R\lim \tau_{\geq -n} K$ by Lemma \ref{lemma-is-limit-dimension} with $d = 0$. Let $U \in \mathcal{B}$. By Equation (\ref{equation-ses-Rlim-over-U}) we get a short exact sequence $$ 0 \to R^1\lim H^{q - 1}(U, \tau_{\geq -n}K) \to H^q(U, K) \to \lim H^q(U, \tau_{\geq -n}K) \to 0 $$ Condition (2) implies $H^q(U, \tau_{\geq -n} K) = H^0(U, H^q(\tau_{\geq -n} K))$ for all $q$ by using the spectral sequence of Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}. The spectral sequence converges because $\tau_{\geq -n}K$ is bounded below. If $n > -q$ then we have $H^q(\tau_{\geq -n}K) = H^q(K)$. Thus the systems on the left and the right of the displayed short exact sequence are eventually constant with values $H^0(U, H^{q - 1}(K))$ and $H^0(U, H^q(K))$ and the lemma follows.