hcju/stacksqa huggingface dataset:
Let $U \in \mathcal{B}$. Note that $H^m(\mathcal{F}^\bullet(U))$ is the cohomology of $$ \lim_n \mathcal{F}_n^{m - 2}(U) \to \lim_n \mathcal{F}_n^{m - 1}(U) \to \lim_n \mathcal{F}_n^m(U) \to \lim_n \mathcal{F}_n^{m + 1}(U) $$ in the third spot from the left. By assumptions (2)(a) and (2)(b) we may apply More on Algebra, Lemma \ref{more-algebra-lemma-apply-Mittag-Leffler-again} to conclude that $$ H^m(\mathcal{F}^\bullet(U)) = \lim H^m(\mathcal{F}_n^\bullet(U)) $$ By assumption (2)(c) we conclude $$ H^m(\mathcal{F}^\bullet(U)) = H^m(\mathcal{F}_n^\bullet(U)) $$ for all $n \geq n_0$. By assumption (1) we conclude that the sheafification of $U \mapsto H^m(\mathcal{F}^\bullet(U))$ is equal to the sheafification of $U \mapsto H^m(\mathcal{F}_n^\bullet(U))$ for all $n \geq n_0$. Thus the inverse system of sheaves $H^m(\mathcal{F}_n^\bullet)$ is constant for $n \geq n_0$ with value $H^m(\mathcal{F}^\bullet)$ which proves the lemma.
Let $V \in \mathcal{B}$. Let $\{V_i \to V\}$ be in the set $\text{Cov}_V$ of Situation \ref{situation-olsson-laszlo}. Because $K_n$ is bounded below there is a spectral sequence $$ E_2^{p, q} = H^p(V_i, H^q(K_n)) $$ converging to $H^{p + q}(V_i, K_n)$. See Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}. Observe that $E_2^{p, q} = 0$ for $p > d_V$ by assumption. Pick $n_0$ such that $$ \begin{matrix} \mathcal{H}^{j + 1} & = & H^{j + 1}(K_n), \\ \mathcal{H}^j & = & H^j(K_n), \\ \ldots, \\ \mathcal{H}^{j - d_V - 2} & = & H^{j - d_V - 2}(K_n) \end{matrix} $$ for all $n \geq n_0$. Comparing the spectral sequences above for $K_n$ and $K_{n_0}$, we see that for $n \geq n_0$ the cohomology groups $H^{j - 1}(V_i, K_n)$ and $H^j(V_i, K_n)$ are independent of $n$. It follows that the map on sections $H^j(R\lim K_n)(V) \to H^j(K_n)(V)$ is injective for $n$ large enough (depending on $V$), see Lemma \ref{lemma-cohomology-derived-limit-injective}. Since every object of $\mathcal{C}$ can be covered by elements of $\mathcal{B}$, we conclude that the map $H^j(R\lim K_n) \to \mathcal{H}^j$ is injective. \medskip\noindent Surjectivity is shown in a similar manner. Namely, pick $U \in \Ob(\mathcal{C})$ and $\gamma \in \mathcal{H}^j(U)$. We want to lift $\gamma$ to a section of $H^j(R\lim K_n)$ after replacing $U$ by the members of a covering. Hence we may assume $U = V \in \mathcal{B}$ by property (1) of Situation \ref{situation-olsson-laszlo}. Pick $n_0$ such that $$ \begin{matrix} \mathcal{H}^{j + 1} & = & H^{j + 1}(K_n), \\ \mathcal{H}^j & = & H^j(K_n), \\ \ldots, \\ \mathcal{H}^{j - d_V - 2} & = & H^{j - d_V - 2}(K_n) \end{matrix} $$ for all $n \geq n_0$. Choose an element $\{V_i \to V\}$ of $\text{Cov}_V$ such that $\gamma|_{V_i} \in \mathcal{H}^j(V_i) = H^j(K_{n_0})(V_i)$ lifts to an element $\gamma_{n_0, i} \in H^j(V_i, K_{n_0})$. This is possible because $H^j(K_{n_0})$ is the sheafification of $U \mapsto H^j(U, K_{n_0})$ by Lemma \ref{lemma-sheafification-cohomology}. By the discussion in the first paragraph of the proof we have that $H^{j - 1}(V_i, K_n)$ and $H^j(V_i, K_n)$ are independent of $n \geq n_0$. Hence $\gamma_{n_0, i}$ lifts to an element $\gamma_i \in H^j(V_i, R\lim K_n)$ by Lemma \ref{lemma-RGamma-commutes-with-Rlim}. This finishes the proof.
By Lemma \ref{lemma-olsson-laszlo} we have $K = R\lim \tau_{\geq -n}K$. By Lemma \ref{lemma-Rf-commutes-with-Rlim} we have $Rf_*K = R\lim Rf_*\tau_{\geq -n}K$. The complexes $Rf_*\tau_{\geq -n}K$ are bounded below. The spectral sequence $$ E_2^{p, q} = R^pf_*H^q(\tau_{\geq -n}K) $$ converging to $H^{p + q}(Rf_*\tau_{\geq -n}K)$ (Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}) and assumption (3) show that $Rf_*\tau_{\geq -n}K$ lies in $D^+_{\mathcal{A}'}(\mathcal{O}')$, see Homology, Lemma \ref{homology-lemma-biregular-ss-converges}. Observe that for $m \geq n$ the map $$ Rf_*(\tau_{\geq -m}K) \longrightarrow Rf_*(\tau_{\geq -n}K) $$ induces an isomorphism on cohomology sheaves in degrees $j \geq -n + N$ by the spectral sequences above. Hence we may apply Lemma \ref{lemma-olsson-laszlo-modified} to conclude.
Let $K$ be in $D_\mathcal{A}(\mathcal{O})$. By Lemma \ref{lemma-olsson-laszlo} we have $K = R\lim \tau_{\geq -n}K$. By Lemma \ref{lemma-Rf-commutes-with-Rlim} we have $Rf_*K = R\lim Rf_*(\tau_{\geq -n}K)$. Let $V' \in \mathcal{B}'$ and let $\{V'_i \to V'\}$ be an element of $\text{Cov}_{V'}$. Then we consider $$ H^j(V'_i, Rf_*K) = H^j(u(V'_i), K) \quad\text{and}\quad H^j(V'_i, Rf_*(\tau_{\geq -n}K)) = H^j(u(V'_i), \tau_{\geq -n}K) $$ The assumption in Situation \ref{situation-olsson-laszlo-prime} implies that the last group is independent of $n$ for $n$ large enough depending on $j$ and $d_{V'}$. Some details omitted. We apply this for $j$ and $j - 1$ and via Lemma \ref{lemma-RGamma-commutes-with-Rlim} this gives that $$ H^j(V'_i, Rf_*K) = \lim H^j(V'_i, Rf_*(\tau_{\geq -n} K)) $$ and the system on the right is constant for $n$ larger than a constant depending only on $d_{V'}$ and $j$. Thus Lemma \ref{lemma-cohomology-derived-limit-injective} implies that $$ H^j(Rf_*K)(V') \longrightarrow \left(\lim H^j(Rf_*(\tau_{\geq -n}K))\right)(V') $$ is injective. Since the elements $V' \in \mathcal{B}'$ cover every object of $\mathcal{C}'$ we conclude that the map $H^j(Rf_*K) \to \lim H^j(Rf_*(\tau_{\geq -n}K))$ is injective. The spectral sequence $$ E_2^{p, q} = R^pf_*H^q(\tau_{\geq -n}K) $$ converging to $H^{p + q}(Rf_*(\tau_{\geq -n}K))$ (Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}) and assumption (3) show that $H^j(Rf_*(\tau_{\geq -n}K))$ is constant for $n \geq N - j$. Hence $H^j(Rf_*K) \to H^j(Rf_*(\tau_{\geq -n}K))$ is injective for $j \geq N - n$. \medskip\noindent Thus we proved the lemma with ``isomorphism'' in the last line of the lemma replaced by ``injective''. However, now choose $j$ and $n$ with $j \geq N - n$. Then consider the distinguished triangle $$ \tau_{\leq -n - 1}K \to K \to \tau_{\geq -n}K \to (\tau_{\leq -n - 1}K)[1] $$ See Derived Categories, Remark \ref{derived-remark-truncation-distinguished-triangle}. Since $\tau_{\geq -n}\tau_{\leq -n -1}K = 0$, the injectivity already proven for $\tau_{-n - 1}K$ implies $$ 0 = H^j(Rf_*(\tau_{\leq -n - 1}K)) = H^{j + 1}(Rf_*(\tau_{\leq -n - 1}K)) = H^{j + 2}(Rf_*(\tau_{\leq -n - 1}K)) = \ldots $$ By the long exact cohomology sequence associated to the distinguished triangle $$ Rf_*(\tau_{\leq -n - 1}K) \to Rf_*K \to Rf_*(\tau_{\geq -n}K) \to Rf_*(\tau_{\leq -n - 1}K)[1] $$ this implies that $H^j(Rf_*K) \to H^j(Rf_*(\tau_{\geq -n}K))$ is an isomorphism.
By rotating the triangle we may assume $c^{K_1}_{X, Z, Y, E}$ and $c^{K_2}_{X, Z, Y, E}$ are quasi-isomorphisms. Choose a map $f : \mathcal{I}^\bullet_1 \to \mathcal{I}^\bullet_2$ of K-injective complexes of $\mathcal{O}$-modules representing $K_1 \to K_2$. Then $K_3$ is represented by the K-injective complex $C(f)^\bullet$, see Derived Categories, Lemma \ref{derived-lemma-triangle-K-injective}. Then the morphism $c^{K_3}_{X, Z, Y, E}$ is an isomorphism as it is the third leg in a map of distinguished triangles in $K(\textit{Ab})$ whose other two legs are quasi-isomorphisms. Some details omitted; use Derived Categories, Lemma \ref{derived-lemma-third-isomorphism-triangle}.
Since sheafification is exact and since $R\Gamma(\mathcal{G}, -) = R\Gamma(\mathcal{G}^\#, -)$ we may assume $\mathcal{D}, \mathcal{E}, \mathcal{F}, \mathcal{G}$ are sheaves of sets. Moreover, the cohomology $R\Gamma(\mathcal{G}, -)$ only depends on the topos, not on the underlying site. Hence by Sites, Lemma \ref{sites-lemma-topos-good-site} we may replace $\mathcal{C}$ by a ``larger'' site with a subcanonical topology such that $\mathcal{G} = h_X$, $\mathcal{F} = h_Y$, $\mathcal{E} = h_Z$, and $\mathcal{D} = h_E$ for some objects $X, Y, Z, E$ of $\mathcal{C}$. In this case the result follows from Lemma \ref{lemma-square-triangle}.
See Derived Categories, Definition \ref{derived-definition-saturated} for the definition of saturated in this setting. See Derived Categories, Lemma \ref{derived-lemma-triangulated-subcategory} for a discussion of triangulated subcategories. The canonical map of the lemma is the unit of the adjoint pair of functors $(Lf^*, Rf_*)$, see Lemma \ref{lemma-adjoint}. Having said this the proof that $D'$ is a saturated triangulated subcategory is omitted; it follows formally from the fact that $Lf^*$ and $Rf_*$ are exact functors of triangulated categories. The final part follows formally from fact that $Lf^*$ and $Rf_*$ are adjoint; compare with Categories, Lemma \ref{categories-lemma-adjoint-fully-faithful}.
See Derived Categories, Definition \ref{derived-definition-saturated} for the definition of saturated in this setting. See Derived Categories, Lemma \ref{derived-lemma-triangulated-subcategory} for a discussion of triangulated subcategories. The canonical map of the lemma is the counit of the adjoint pair of functors $(Lf^*, Rf_*)$, see Lemma \ref{lemma-adjoint}. Having said this the proof that $D'$ is a saturated triangulated subcategory is omitted; it follows formally from the fact that $Lf^*$ and $Rf_*$ are exact functors of triangulated categories. The final part follows formally from fact that $Lf^*$ and $Rf_*$ are adjoint; compare with Categories, Lemma \ref{categories-lemma-adjoint-fully-faithful}.
Observe that $f^*Rf_*K \to K$ is an isomorphism if and only if it is an isomorphism on cohomology sheaves $H^j$. Observe that $H^j(f^*Rf_*K) = f^*H^j(Rf_*K) = f^*H^j(Rf_*\tau_{\leq j}K) = H^j(f^*Rf_*\tau_{\leq j}K)$. Hence we may assume that $K$ is bounded. Then property (3) tells us the cohomology sheaves are in the triangulated subcategory $D' \subset D(\mathcal{O}_\mathcal{C})$ of Lemma \ref{lemma-upstairs}. Hence $K$ is in it too.
Observe that $K \to Rf_*f^*K$ is an isomorphism if and only if it is an isomorphism on cohomology sheaves $H^j$. Observe that $H^j(Rf_*f^*K) = H^j(Rf_*\tau_{\leq j}f^*K) = H^j(Rf_*f^*\tau_{\leq j}K)$. Hence we may assume that $K$ is bounded. Then property (3) tells us the cohomology sheaves are in the triangulated subcategory $D' \subset D(\mathcal{O}_\mathcal{D})$ of Lemma \ref{lemma-downstairs}. Hence $K$ is in it too.
By assumptions (2) and (3) and Lemmas \ref{lemma-bounded-in-image-upstairs} and \ref{lemma-downstairs} we see that $f^* : D_{\mathcal{A}'}^+(\mathcal{O}') \to D_\mathcal{A}^+(\mathcal{O})$ is fully faithful. Let $\mathcal{F} \in \Ob(\mathcal{A})$. Then we can write $\mathcal{F} = f^*\mathcal{F}'$. Then $Rf_*\mathcal{F} = Rf_* f^*\mathcal{F}' = \mathcal{F}'$. In particular, we have $R^pf_*\mathcal{F} = 0$ for $p > 0$ and $f_*\mathcal{F} \in \Ob(\mathcal{A}')$. Thus for any $K \in D^+_\mathcal{A}(\mathcal{O})$ we see, using the spectral sequence $E_2^{p, q} = R^pf_*H^q(K)$ converging to $R^{p + q}f_*K$, that $Rf_*K$ is in $D^+_{\mathcal{A}'}(\mathcal{O}')$. Of course, it also follows from Lemmas \ref{lemma-bounded-in-image-downstairs} and \ref{lemma-upstairs} that $Rf_* : D_\mathcal{A}^+(\mathcal{O}) \to D_{\mathcal{A}'}^+(\mathcal{O}')$ is fully faithful. Since $f^*$ and $Rf_*$ are adjoint we then get the result of the lemma, for example by Categories, Lemma \ref{categories-lemma-adjoint-fully-faithful}.
Since $f^*$ is exact, it is clear that $f^*$ defines a functor $f^* : D_{\mathcal{A}'}(\mathcal{O}') \to D_\mathcal{A}(\mathcal{O})$ as in the statement of the lemma and that moreover this functor commutes with the truncation functors $\tau_{\geq -n}$. We already know that $f^*$ and $Rf_*$ are quasi-inverse equivalence on the corresponding bounded below categories, see Lemma \ref{lemma-equivalence-bounded}. By Lemma \ref{lemma-olsson-laszlo-map-version-one} with $N = 0$ we see that $Rf_*$ indeed defines a functor $Rf_* : D_\mathcal{A}(\mathcal{O}) \to D_{\mathcal{A}'}(\mathcal{O}')$ and that moreover this functor commutes with the truncation functors $\tau_{\geq -n}$. Thus for $K$ in $D_\mathcal{A}(\mathcal{O})$ the map $f^*Rf_*K \to K$ is an isomorphism as this is true on trunctions. Similarly, for $K'$ in $D_{\mathcal{A}'}(\mathcal{O}')$ the map $K' \to Rf_*f^*K'$ is an isomorphism as this is true on trunctions. This finishes the proof.
This helper lemma is an almost immediate consequence of Lemma \ref{lemma-square-triangle} and we strongly urge the reader skip the proof. Say $K' = R\epsilon_*K$. Choose a K-injective complex of $\mathcal{O}_\tau$-modules $\mathcal{J}^\bullet$ representing $K$. Then $\epsilon_*\mathcal{J}^\bullet$ is a K-injective complex of $\mathcal{O}_{\tau'}$-modules representing $K'$, see Lemma \ref{lemma-K-injective-flat}. Next, $$ 0 \to \mathcal{J}^\bullet(X) \xrightarrow{\alpha} \mathcal{J}^\bullet(Z) \oplus \mathcal{J}^\bullet(Y) \xrightarrow{\beta} \mathcal{J}^\bullet(E) \to 0 $$ is a short exact sequence of complexes of abelian groups, see Lemma \ref{lemma-square-triangle} and its proof. Since this is the same as the sequence of complexes of abelian groups which is used to define $c^{K'}_{X, Z, Y, E}$, we conclude.
By Lemma \ref{lemma-Leray-unbounded} we have $H^n_\tau(X, \epsilon_X^{-1}L) = H^n_{\tau'}(X, R\epsilon_{X, *}\epsilon_X^{-1}L)$. There is a spectral sequence $$ E_2^{p, q} = R^p\epsilon_{X, *}\epsilon_X^{-1}H^q(L) $$ converging to $H^{p + q}(R\epsilon_{X, *}\epsilon_X^{-1}L)$. By $(V_n)$ we have the vanishing of $E_2^{p, q}$ for $0 < p \leq n$ and $0 \leq q \leq n$. Thus $E_2^{0, q} = \epsilon_{X, *}\epsilon_X^{-1}H^q(L) = H^q(L)$ are the only nonzero terms $E_2^{p, q}$ with $p + q \leq n$. It follows that the map $$ L \longrightarrow R\epsilon_{X, *}\epsilon_X^{-1}L $$ is an isomorphism in degrees $\leq n$ (small detail omitted). Hence we find that $H^i_{\tau'}(X, L) = H^i_{\tau'}(X, R\epsilon_{X, *}\epsilon_X^{-1}L)$ for $i \leq n$. Thus the lemma is proved.
Recall that $\epsilon_X^{-1}\epsilon_{X, *}\mathcal{F} = \mathcal{F}$ hence the map is given by pulling back cohomology classes by $\epsilon_X$. The Leray spectral sequence (Lemma \ref{lemma-Leray}) $$ E_2^{p, q} = H^p_{\tau'}(X, R^q\epsilon_{X, *}\mathcal{F}) \Rightarrow H^{p + q}_\tau(X, \mathcal{F}) $$ combined with the assumed vanishing gives an exact sequence $$ 0 \to H^{n + 1}_{\tau'}(X, \epsilon_{X, *}\mathcal{F}) \to H^{n + 1}_\tau(X, \mathcal{F}) \to H^0_{\tau'}(X, R^{n + 1}\epsilon_{X, *}\mathcal{F}) $$ This is a restatement of the lemma.
Observe that $X \times_Y X$ exists by (\ref{item-base-change-P}). For $Z$ in $\mathcal{C}/Y$ denote $\mathcal{F}'|_Z$ the restriction of $\mathcal{F}'$ to $\mathcal{C}_{\tau'}/Z$. Recall that $H^{n + 1}_{\tau'}(X, \mathcal{F}') = H^{n + 1}(\mathcal{C}_{\tau'}/X, \mathcal{F}'|_X)$, see Lemma \ref{lemma-cohomology-of-open}. The lemma asserts that the image $\overline{\theta} \in H^0(Y, R^{n + 1}f_{\tau', *}\mathcal{F}'|_X)$ of $\theta$ is zero. Consider the cartesian diagram $$ \xymatrix{ X \times_Y X \ar[d]_{\text{pr}_1} \ar[r]_{\text{pr}_2} & X \ar[d]^f \\ X \ar[r]^f & Y } $$ By trivial base change (Lemma \ref{lemma-localize-cartesian-square}) we have $$ f_{\tau'}^{-1}R^{n + 1}f_{\tau', *}(\mathcal{F}'|_X) = R^{n + 1}\text{pr}_{1, \tau', *}(\mathcal{F}'|_{X \times_Y X}) $$ If $\text{pr}_1^{-1}\theta = \text{pr}_2^{-1}\theta$, then the section $f_{\tau'}^{-1}\overline{\theta}$ of $f_{\tau'}^{-1}R^{n + 1}f_{\tau', *}(\mathcal{F}'|_X)$ is zero, because it is clear that $\text{pr}_1^{-1}\theta$ maps to the zero element in $H^0(X, R^{n + 1}\text{pr}_{1, \tau', *}(\mathcal{F}'|_{X \times_Y X}))$. By (\ref{item-restriction-A}) we have $\mathcal{F}'|_X$ in $\mathcal{A}'_X$. Thus $\mathcal{G}' = R^{n + 1}f_{\tau', *}(\mathcal{F}'|_X)$ is an object of $\mathcal{A}'_Y$ by (\ref{item-A-and-P}). Thus $\mathcal{G}'$ satisfies the sheaf property for $\tau$-coverings by (\ref{item-A-sheaf}). Since $\{X \to Y\}$ is a $\tau$-covering we conclude that restriction $\mathcal{G}'(Y) \to \mathcal{G}'(X)$ is injective. It follows that $\overline{\theta}$ is zero.
Observe that $(V_0)$ holds as it is the empty condition. Then we get $(V_n)$ for all $n$ by Lemma \ref{lemma-induction-step-V-C-general}. \medskip\noindent Proof of (1). The object $K = \epsilon_X^{-1}K'$ has cohomology sheaves $H^i(K) = \epsilon_X^{-1}H^i(K')$ in $\mathcal{A}_X$. Hence the spectral sequence $$ E_2^{p, q} = R^p\epsilon_{X, *} H^q(K) \Rightarrow H^{p + q}(R\epsilon_{X, *}K) $$ degenerates by $(V_n)$ for all $n$ and we find $$ H^n(R\epsilon_{X, *}K) = \epsilon_{X, *}H^n(K) = \epsilon_{X, *}\epsilon_X^{-1}H^i(K') = H^i(K'). $$ again because $H^i(K')$ is in $\mathcal{A}'_X$. Thus the canonical map $K' \to R\epsilon_{X, *}(\epsilon_X^{-1}K')$ is an isomorphism. \medskip\noindent Proof of (2). Using the spectral sequence $$ E_2^{p, q} = R^pf_{\tau', *}H^q(K') \Rightarrow R^{p + q}f_{\tau', *}K' $$ the fact that $R^pf_{\tau', *}H^q(K')$ is in $\mathcal{A}'_Y$ by (\ref{item-A-and-P}), the fact that $\mathcal{A}'_Y$ is a weak Serre subcategory of $\textit{Ab}(\mathcal{C}_{\tau'}/Y)$, and Homology, Lemma \ref{homology-lemma-biregular-ss-converges} we conclude that $Rf_{\tau', *}K' \in D^+_{\mathcal{A}'_X}(\mathcal{C}_{\tau'}/X)$. To finish the proof we have to show the base change map $$ \epsilon_Y^{-1}(Rf_{\tau', *}K') \longrightarrow Rf_{\tau, *}(\epsilon_X^{-1}K') $$ is an isomorphism. Comparing the spectral sequence above to the spectral sequence $$ E_2^{p, q} = R^pf_{\tau, *}H^q(\epsilon_X^{-1}K') \Rightarrow R^{p + q}f_{\tau, *}\epsilon_X^{-1}K' $$ we reduce this to the case where $K'$ has a single nonzero cohomology sheaf $\mathcal{F}'$ in $\mathcal{A}'_X$; details omitted. Then Lemma \ref{lemma-V-implies-C-general} gives $\epsilon_Y^{-1}R^if_{\tau', *}\mathcal{F}' = R^if_{\tau, *}\epsilon_X^{-1}\mathcal{F}'$ for all $i$ and the proof is complete.
We need to check the conditions listed in Homology, Lemma \ref{homology-lemma-characterize-weak-serre-subcategory} for $\mathcal{A}_X$. If $\varphi : \mathcal{F} \to \mathcal{G}$ is a map in $\mathcal{A}_X$, then $\epsilon_{X, *}\varphi : \epsilon_{X, *}\mathcal{F} \to \epsilon_{X, *}\mathcal{G}$ is a map in $\mathcal{A}'_X$. Hence $\Ker(\epsilon_{X, *}\varphi)$ and $\Coker(\epsilon_{X, *}\varphi)$ are objects of $\mathcal{A}'_X$ as this is a weak Serre subcategory of $\textit{Ab}(\mathcal{C}_{\tau'}/X)$. Applying $\epsilon_X^{-1}$ we obtain an exact sequence $$ 0 \to \epsilon_X^{-1}\Ker(\epsilon_{X, *}\varphi) \to \mathcal{F} \to \mathcal{G} \to \epsilon_X^{-1}\Coker(\epsilon_{X, *}\varphi) \to 0 $$ and we see that $\Ker(\varphi)$ and $\Coker(\varphi)$ are in $\mathcal{A}_X$. Finally, suppose that $$ 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 $$ is a short exact sequence in $\textit{Ab}(\mathcal{C}_\tau/X)$ with $\mathcal{F}_1$ and $\mathcal{F}_3$ in $\mathcal{A}_X$. Then applying $\epsilon_{X, *}$ we obtain an exact sequence $$ 0 \to  \epsilon_{X, *}\mathcal{F}_1 \to \epsilon_{X, *}\mathcal{F}_2 \to \epsilon_{X, *}\mathcal{F}_3 \to R^1\epsilon_{X, *}\mathcal{F}_1 = 0 $$ Vanishing by Lemma \ref{lemma-V-C-all-n-general}. Hence $\epsilon_{X, *}\mathcal{F}_2$ is in $\mathcal{A}'_X$ as this is a weak Serre subcategory of $\textit{Ab}(\mathcal{C}_{\tau'}/X)$. Pulling back by $\epsilon_X$ we conclude that $\mathcal{F}_2$ is in $\mathcal{A}_X$. \medskip\noindent Thus $\mathcal{A}_X$ is a weak Serre subcategory of $\textit{Ab}(\mathcal{C}_\tau/X)$ and it makes sense to consider the category $D^+_{\mathcal{A}_X}(\mathcal{C}_\tau/X)$. Observe that $\epsilon_X^{-1} : \mathcal{A}'_X \to \mathcal{A}_X$ is an equivalence and that $\mathcal{F}' \to R\epsilon_{X, *}\epsilon_X^{-1}\mathcal{F}'$ is an isomorphism for $\mathcal{F}'$ in $\mathcal{A}'_X$ since we have $(V_n)$ for all $n$ by Lemma \ref{lemma-V-C-all-n-general}. Thus we conclude by Lemma \ref{lemma-equivalence-bounded}.
The final object is the singleton space. Given morphisms $X \to Z$ and $Y \to Z$ of $\textit{LC}$ the fibre product $X \times_Z Y$ is a subspace of $X \times Y$. Hence $X \times_Z Y$ is Hausdorff as $X \times Y$ is Hausdorff by Topology, Section \ref{topology-section-Hausdorff}. \medskip\noindent If $X$ and $Y$ are quasi-compact, then $X \times Y$ is quasi-compact by  Topology, Theorem \ref{topology-theorem-tychonov}. Since $X \times_Z Y$ is a closed subset of $X \times Y$ (Topology, Lemma \ref{topology-lemma-fibre-product-closed}) we find that $X \times_Z Y$ is quasi-compact by Topology, Lemma \ref{topology-lemma-closed-in-quasi-compact}. \medskip\noindent Finally, returning to the general case, if $x \in X$ and $y \in Y$ we can pick quasi-compact neighbourhoods $x \in E \subset X$ and $y \in F \subset Y$ and we find that $E \times_Z F$ is a quasi-compact neighbourhood of $(x, y)$ by the result above. Thus $X \times_Z Y$ is an object of $\textit{LC}$ by Topology, Lemma \ref{topology-lemma-locally-quasi-compact-Hausdorff}.
Let $y \in Y$ be a point. For each $x \in X_y$ choose a quasi-compact neighbourhood $E_x \subset X$. Choose $x \in U_x \subset E_x$ open. Since $f$ is proper the fibre $X_y$ is quasi-compact and we find $x_1, \ldots, x_n \in X_y$ such that $X_y \subset U_{x_1} \cup \ldots \cup U_{x_n}$. We claim that $f(E_{x_1}) \cup \ldots \cup f(E_{x_n})$ is a neighbourhood of $y$. Namely, as $f$ is closed (Topology, Theorem \ref{topology-theorem-characterize-proper}) we see that $Z = f(X \setminus U_{x_1} \cup \ldots \cup U_{x_n})$ is a closed subset of $Y$ not containing $y$. As $f$ is surjective we see that $Y \setminus Z$ is contained in $f(E_{x_1}) \cup \ldots \cup f(E_{x_n})$ as desired.
The morphism of topoi $f_{qc}$ is the one from Sites, Lemma \ref{sites-lemma-relocalize} which in our case comes from the continuous functor $Z/Y \mapsto Z \times_Y X/X$, see Sites, Lemma \ref{sites-lemma-relocalize-given-fibre-products}. The diagram on the left commutes because the corresponding continuous functors compose correctly (see Sites, Lemma \ref{sites-lemma-composition-morphisms-sites}). The diagram on the right commutes because the one on the left does and because of part (5) of Lemma \ref{lemma-collect-true-things-Zar}. \medskip\noindent Proof of the final assertion. The reader may repeat the proof of part (7a) of Lemma \ref{lemma-collect-true-things-Zar}; we will instead deduce this from it. As $\epsilon_{Y, *}$ is the identity functor on underlying presheaves, it reflects isomorphisms. The description in Lemma \ref{lemma-describe-pullback-pi} shows that $\epsilon_{Y, *} \circ a_Y^{-1} = \pi_Y^{-1}$ and similarly for $X$. To show that the canonical map $a_Y^{-1}f_*\mathcal{F} \to f_{qc, *}a_X^{-1}\mathcal{F}$ is an isomorphism, it suffices to show that $$ \pi_Y^{-1}f_*\mathcal{F} = \epsilon_{Y, *}a_Y^{-1}f_*\mathcal{F} \to \epsilon_{Y, *}f_{qc, *}a_X^{-1}\mathcal{F} = f_{Zar, *}\epsilon_{X, *}a_X^{-1}\mathcal{F} = f_{Zar, *}\pi_X^{-1}\mathcal{F} $$ is an isomorphism. This is part (7a) of Lemma \ref{lemma-collect-true-things-Zar}.