hcju/stacksqa huggingface dataset:
As $L\pi_!$ agrees for modules and abelian sheaves by Lemma \ref{lemma-properties-lower-shriek-fibred-category} it suffices to prove this when $\mathcal{F}$ is an abelian sheaf. For $U \in \Ob(\mathcal{C})$ the abelian sheaf $j_{U!}\mathbf{Z}_U$ is a projective object of $\textit{Ab}(\mathcal{C})$ since $\Hom(j_{U!}\mathbf{Z}_U, \mathcal{F}) = \mathcal{F}(U)$ and taking sections is an exact functor as the topology is chaotic. Every abelian sheaf is a quotient of a direct sum of $j_{U!}\mathbf{Z}_U$ by Modules on Sites, Lemma \ref{sites-modules-lemma-module-quotient-flat}. Thus we can compute $L\pi_!(\mathcal{F})$ by choosing a resolution $$ \ldots \to \mathcal{G}^{-1} \to \mathcal{G}^0 \to \mathcal{F} \to 0 $$ whose terms are direct sums of sheaves of the form above and taking $L\pi_!(\mathcal{F}) = \pi_!(\mathcal{G}^\bullet)$. Consider the double complex $A^{\bullet, \bullet} = \mathcal{G}^\bullet(U_\bullet)$. The map $\mathcal{G}^0 \to \mathcal{F}$ gives a map of complexes $A^{0, \bullet} \to \mathcal{F}(U_\bullet)$. Since $\pi_!$ is computed by taking the colimit over $\mathcal{C}^{opp}$ (Lemma \ref{lemma-compute-pi-shriek}) we see that the two compositions $\mathcal{G}^m(U_1) \to \mathcal{G}^m(U_0) \to \pi_!\mathcal{G}^m$ are equal. Thus we obtain a canonical map of complexes $$ \text{Tot}(A^{\bullet, \bullet}) \longrightarrow \pi_!(\mathcal{G}^\bullet) = L\pi_!(\mathcal{F}) $$ To prove the lemma it suffices to show that the complexes $$ \ldots \to \mathcal{G}^m(U_1) \to \mathcal{G}^m(U_0) \to \pi_!\mathcal{G}^m \to 0 $$ are exact, see Homology, Lemma \ref{homology-lemma-double-complex-gives-resolution}. Since the sheaves $\mathcal{G}^m$ are direct sums of the sheaves $j_{U!}\mathbf{Z}_U$ we reduce to $\mathcal{G} = j_{U!}\mathbf{Z}_U$. The complex $j_{U!}\mathbf{Z}_U(U_\bullet)$ is the complex of abelian groups associated to the free $\mathbf{Z}$-module on the simplicial set $\Mor_\mathcal{C}(U_\bullet, U)$ which we assumed to be homotopy equivalent to a singleton. We conclude that $$ j_{U!}\mathbf{Z}_U(U_\bullet) \to \mathbf{Z} $$ is a homotopy equivalence of abelian groups hence a quasi-isomorphism (Simplicial, Remark \ref{simplicial-remark-homotopy-better} and Lemma \ref{simplicial-lemma-homotopy-s-N}). This finishes the proof since $\pi_!j_{U!}\mathbf{Z}_U = \mathbf{Z}$ as was shown in the proof of Lemma \ref{lemma-properties-lower-shriek-fibred-category}.
As both sides commute with colimits, it suffices to prove this for $K_1 = j_{U!}\underline{B}_U$ and $K_2 = j_{V!}\underline{B}_V$ for $U \in \Ob(\mathcal{C}_1)$ and $V \in \Ob(\mathcal{C}_2)$. See construction of $L\pi_!$ in Lemma \ref{lemma-existence-derived-lower-shriek}. In this case $$ g_1^{-1}K_1 \otimes_{\underline{B}}^\mathbf{L} g_2^{-1}K_2 = g_1^{-1}K_1 \otimes_{\underline{B}} g_2^{-1}K_2 = j_{(U, V)!}\underline{B}_{(U, V)} $$ Verification omitted. Hence the result follows as both the left and the right hand side of the formula of the lemma evaluate to $B$, see construction of $L\pi_!$ in Lemma \ref{lemma-existence-derived-lower-shriek}.
Consider the diagram of categories and functors $$ \xymatrix{ & & \mathcal{C} \\ \mathcal{C} \ar[r]^-u & \mathcal{C} \times \mathcal{C} \ar[rd]^{u_2} \ar[ru]_{u_1} \\ & & \mathcal{C} } $$ where $u$ is the diagonal functor and $u_i$ are the projection functors. This gives morphisms of ringed topoi $g$, $g_1$, $g_2$. For any object $(U_1, U_2)$ of $\mathcal{C}$ we have $$ \Mor_{\mathcal{C} \times \mathcal{C}}(u(U_\bullet), (U_1, U_2)) = \Mor_\mathcal{C}(U_\bullet, U_1) \times \Mor_\mathcal{C}(U_\bullet, U_2) $$ which is homotopy equivalent to a point by Simplicial, Lemma \ref{simplicial-lemma-products-homotopy}. Thus Lemma \ref{lemma-get-it-now} gives $L\pi_!(g^{-1}K) = L(\pi \times \pi)_!(K)$ for any $K$ in $D(\mathcal{C} \times \mathcal{C}, B)$. Take $K = g_1^{-1}K_1 \otimes_B^\mathbf{L} g_2^{-1}K_2$. Then $g^{-1}K = K_1 \otimes^\mathbf{L}_{\underline{B}} K_2$ because $g^{-1} = g^* = Lg^*$ commutes with derived tensor product (Lemma \ref{lemma-pullback-tensor-product}). To finish we apply Lemma \ref{lemma-product-categories}.
Note: in this proof $L\pi_!$ denotes the left derived functor of $\pi_!$ on abelian sheaves. Since $L\pi_!$ commutes with colimits, it suffices to prove this for bounded above complexes of $\mathcal{O}$-modules (compare with argument of Derived Categories, Proposition \ref{derived-proposition-left-derived-exists} or just stick to bounded above complexes). Every such complex is quasi-isomorphic to a bounded above complex whose terms are direct sums of $j_{U!}\mathcal{O}_U$ with $U \in \Ob(\mathcal{C})$, see Modules on Sites, Lemma \ref{sites-modules-lemma-module-quotient-flat}. Thus it suffices to prove the lemma for $j_{U!}\mathcal{O}_U$. By assumption $$ S_\bullet = \Mor_\mathcal{C}(U_\bullet, U) $$ is a simplicial set homotopy equivalent to the constant simplicial set on a singleton. Set $P_n = \mathcal{O}(U_n)$ and $P'_n = \mathcal{O}'(U_n)$. Observe that the complex associated to the simplicial abelian group $$ X_\bullet : n \longmapsto \bigoplus\nolimits_{s \in S_n} P_n $$ computes $L\pi_!(j_{U!}\mathcal{O}_U)$ by Lemma \ref{lemma-compute-by-cosimplicial-resolution}. Since $j_{U!}\mathcal{O}_U$ is a flat $\mathcal{O}$-module we have $j_{U!}\mathcal{O}_U \otimes^\mathbf{L}_\mathcal{O} \mathcal{O}' = j_{U!}\mathcal{O}'_U$ and $L\pi_!$ of this is computed by the complex associated to the simplicial abelian group $$ X'_\bullet : n \longmapsto \bigoplus\nolimits_{s \in S_n} P'_n $$ As the rule which to a simplicial set $T_\bullet$ associated the simplicial abelian group with terms $\bigoplus_{t \in T_n} P_n$ is a functor, we see that $X_\bullet \to P_\bullet$ is a homotopy equivalence of simplicial abelian groups. Similarly, the rule which to a simplicial set $T_\bullet$ associates the simplicial abelian group with terms $\bigoplus_{t \in T_n} P'_n$ is a functor. Hence $X'_\bullet \to P'_\bullet$ is a homotopy equivalence of simplicial abelian groups. By assumption $P_\bullet \to P'_\bullet$ is a quasi-isomorphism (since $P_\bullet$, resp.\ $P'_\bullet$ computes $L\pi_!\mathcal{O}$, resp. $L\pi_!\mathcal{O}'$ by Lemma \ref{lemma-compute-by-cosimplicial-resolution}). We conclude that $X_\bullet$ and $X'_\bullet$ are quasi-isomorphic as desired.
For a morphism $h : V' \to V$ of $\mathcal{D}$ there is a pullback functor $h^* : \mathcal{C}_V \to \mathcal{C}_{V'}$ of fibre categories (Categories, Definition \ref{categories-definition-pullback-functor-fibred-category}). Moreover for $U \in \Ob(\mathcal{C}_V)$ there is a strongly cartesian morphism $h^*U \to U$ covering $h$. Restriction along these strongly cartesian morphisms defines a transformation of functors $$ \mathcal{F}|_{\mathcal{C}_V} \longrightarrow \mathcal{F}|_{\mathcal{C}_{V'}} \circ h^*. $$ By Example \ref{example-morphism-categories} we obtain the desired restriction map $$ H_n(\mathcal{C}_V, \mathcal{F}|_{\mathcal{C}_V}) \longrightarrow H_n(\mathcal{C}_{V'}, \mathcal{F}|_{\mathcal{C}_{V'}}) $$ Let us denote $L_{n, p}(\mathcal{F})$ this presheaf, so that $L_n(\mathcal{F}) = L_{n, p}(\mathcal{F})^\#$. The canonical map $\gamma : \mathcal{F} \to \mathcal{F}^+$ (Sites, Theorem \ref{sites-theorem-plus}) defines a canonical map $L_{n, p}(\mathcal{F}) \to L_{n, p}(\mathcal{F}^+)$. We have to prove this map becomes an isomorphism after sheafification. \medskip\noindent Let us use the computation of homology given in Example \ref{example-left-derived-colimits}. Denote $K_\bullet(\mathcal{F}|_{\mathcal{C}_V})$ the complex associated to the restriction of $\mathcal{F}$ to the fibre category $\mathcal{C}_V$. By the remarks above we obtain a presheaf $K_\bullet(\mathcal{F})$ of complexes $$ V \longmapsto K_\bullet(\mathcal{F}|_{\mathcal{C}_V}) $$ whose cohomology presheaves are the presheaves $L_{n, p}(\mathcal{F})$. Thus it suffices to show that $$ K_\bullet(\mathcal{F}) \longrightarrow K_\bullet(\mathcal{F}^+) $$ becomes an isomorphism on sheafification. \medskip\noindent Injectivity. Let $V$ be an object of $\mathcal{D}$ and let $\xi \in K_n(\mathcal{F})(V)$ be an element which maps to zero in $K_n(\mathcal{F}^+)(V)$. We have to show there exists a covering $\{V_j \to V\}$ such that $\xi|_{V_j}$ is zero in $K_n(\mathcal{F})(V_j)$. We write $$ \xi = \sum (U_{i, n + 1} \to \ldots \to U_{i, 0}, \sigma_i) $$ with $\sigma_i \in \mathcal{F}(U_{i, 0})$. We arrange it so that each sequence of morphisms $U_n \to \ldots \to U_0$ of $\mathcal{C}_V$ occurs are most once. Since the sums in the definition of the complex $K_\bullet$ are direct sums, the only way this can map to zero in $K_\bullet(\mathcal{F}^+)(V)$ is if all $\sigma_i$ map to zero in $\mathcal{F}^+(U_{i, 0})$. By construction of $\mathcal{F}^+$ there exist coverings $\{U_{i, 0, j} \to U_{i, 0}\}$ such that $\sigma_i|_{U_{i, 0, j}}$ is zero. By our construction of the topology on $\mathcal{C}$ we can write $U_{i, 0, j} \to U_{i, 0}$ as the pullback (Categories, Definition \ref{categories-definition-pullback-functor-fibred-category}) of some morphisms $V_{i, j} \to V$ and moreover each $\{V_{i, j} \to V\}$ is a covering. Choose a covering $\{V_j \to V\}$ dominating each of the coverings $\{V_{i, j} \to V\}$. Then it is clear that $\xi|_{V_j} = 0$. \medskip\noindent Surjectivity. Proof omitted. Hint: Argue as in the proof of injectivity.
Consider the sequence of functors $\mathcal{F} \mapsto L_n(\mathcal{F})$ from $\textit{PAb}(\mathcal{C}) \to \textit{Ab}(\mathcal{C})$. Since for each $V \in \Ob(\mathcal{D})$ the sequence of functors $H_n(\mathcal{C}_V, - )$ forms a $\delta$-functor so do the functors $\mathcal{F} \mapsto L_n(\mathcal{F})$. Our goal is to show these form a universal $\delta$-functor. In order to do this we construct some abelian presheaves on which these functors vanish. \medskip\noindent For $U' \in \Ob(\mathcal{C})$ consider the abelian presheaf $\mathcal{F}_{U'} = j_{U'!}^{\textit{PAb}}\mathbf{Z}_{U'}$ (Modules on Sites, Remark \ref{sites-modules-remark-localize-presheaves}). Recall that $$ \mathcal{F}_{U'}(U) = \bigoplus\nolimits_{\Mor_\mathcal{C}(U, U')} \mathbf{Z} $$ If $U$ lies over $V = p(U)$ in $\mathcal{D})$ and $U'$ lies over $V' = p(U')$ then any morphism $a : U \to U'$ factors uniquely as $U \to h^*U' \to U'$ where $h = p(a) : V \to V'$ (see Categories, Definition \ref{categories-definition-pullback-functor-fibred-category}). Hence we see that $$ \mathcal{F}_{U'}|_{\mathcal{C}_V} = \bigoplus\nolimits_{h \in \Mor_\mathcal{D}(V, V')} j_{h^*U'!}\mathbf{Z}_{h^*U'} $$ where $j_{h^*U'} : \Sh(\mathcal{C}_V/h^*U') \to \Sh(\mathcal{C}_V)$ is the localization morphism. The sheaves $j_{h^*U'!}\mathbf{Z}_{h^*U'}$ have vanishing higher homology groups (see Example \ref{example-left-derived-colimits}). We conclude that $L_n(\mathcal{F}_{U'}) = 0$ for all $n > 0$ and all $U'$. It follows that any abelian presheaf $\mathcal{F}$ is a quotient of an abelian presheaf $\mathcal{G}$ with $L_n(\mathcal{G}) = 0$ for all $n > 0$ (Modules on Sites, Lemma \ref{sites-modules-lemma-module-quotient-flat}). Since $L_n(\mathcal{F}) = L_n(\mathcal{F}^\#)$ we see that the same thing is true for abelian sheaves. Thus the sequence of functors $L_n(-)$ is a universal delta functor on $\textit{Ab}(\mathcal{C})$ (Homology, Lemma \ref{homology-lemma-efface-implies-universal}). Since we have agreement with $H^{-n}(L\pi_!(-))$ for $n = 0$ by Lemma \ref{lemma-compute-pi-shriek} we conclude by uniqueness of universal $\delta$-functors (Homology, Lemma \ref{homology-lemma-uniqueness-universal-delta-functor}) and Derived Categories, Lemma \ref{derived-lemma-right-derived-delta-functor}.
Say $p(U) = V$. The category $\mathcal{I}_U$ is the category of pairs $(U', \varphi)$ where $\varphi : U \to u(U')$ is a morphism of $\mathcal{C}$ with $p(\varphi) = \text{id}_V$, i.e., $\varphi$ is a morphism of the fibre category $\mathcal{C}_V$. Morphisms $(U'_1, \varphi_1) \to (U'_2, \varphi_2)$ are given by morphisms $a : U'_1 \to U'_2$ of the fibre category $\mathcal{C}'_V$ such that $\varphi_2 = u(a) \circ \varphi_1$. The presheaf $\mathcal{F}'_U$ sends $(U', \varphi)$ to $\mathcal{F}'(U')$. We will construct the restriction mappings below. \medskip\noindent Choose a factorization $$ \xymatrix{ \mathcal{C}' \ar@<1ex>[r]^{u'} & \mathcal{C}'' \ar[r]^{u''} \ar@<1ex>[l]^w & \mathcal{C} } $$ of $u$ as in Categories, Lemma \ref{categories-lemma-ameliorate-morphism-fibred-categories}. Then $g_! = g''_! \circ g'_!$ and similarly for derived functors. On the other hand, the functor $g'_!$ is exact, see Modules on Sites, Lemma \ref{sites-modules-lemma-have-left-adjoint}. Thus we get $Lg_!(\mathcal{F}') = Lg''_!(\mathcal{F}'')$ where $\mathcal{F}'' = g'_!\mathcal{F}'$. Note that $\mathcal{F}'' = h^{-1}\mathcal{F}'$ where $h : \Sh(\mathcal{C}'') \to \Sh(\mathcal{C}')$ is the morphism of topoi associated to $w$, see Sites, Lemma \ref{sites-lemma-have-left-adjoint}. The functor $u''$ turns $\mathcal{C}''$ into a fibred category over $\mathcal{C}$, hence Lemma \ref{lemma-compute-left-derived-pi-shriek} applies to the computation of $L_ng''_!$. The result follows as the construction of $\mathcal{C}''$ in the proof of Categories, Lemma \ref{categories-lemma-ameliorate-morphism-fibred-categories} shows that the fibre category $\mathcal{C}''_U$ is equal to $\mathcal{I}_U$. Moreover, $h^{-1}\mathcal{F}'|_{\mathcal{C}''_U}$ is given by the rule described above (as $w$ is continuous and cocontinuous by Stacks, Lemma \ref{stacks-lemma-topology-inherited-functorial} so we may apply Sites, Lemma \ref{sites-lemma-when-shriek}).
Let $([n], U)$ be an object of $\Delta \times \mathcal{C}$. Since $L\pi_!$ commutes with colimits, it suffices to prove this for bounded above complexes of $\mathcal{O}$-modules (compare with argument of Derived Categories, Proposition \ref{derived-proposition-left-derived-exists} or just stick to bounded above complexes). Every such complex is quasi-isomorphic to a bounded above complex whose terms are flat modules, see Modules on Sites, Lemma \ref{sites-modules-lemma-module-quotient-flat}. Thus it suffices to prove the lemma for a flat $\mathcal{A}_\bullet$-module $\mathcal{F}$. In this case the derived tensor product is the usual tensor product and is a sheaf also. Hence by Lemma \ref{lemma-compute-left-derived-pi-shriek} we can compute the cohomology sheaves of both sides of the equation by the procedure of Lemma \ref{lemma-compute-left-derived-pi-shriek-pre}. Thus it suffices to prove the result for the restriction of $\mathcal{F}$ to the fibre categories (i.e., to $\Delta \times U$). In this case the result follows from Lemma \ref{lemma-O-homology-qis}.
The first statement follows from the second, hence we only prove (2). We will prove this by induction on the length of the complex $\mathcal{E}^\bullet$. If $\mathcal{E}^\bullet \cong \mathcal{E}[-n]$ for some direct summand $\mathcal{E}$ of a finite free $\mathcal{O}$-module and integer $n \geq a$, then the result follows from Lemma \ref{lemma-local-lift-map} and the fact that $\mathcal{F}^{n - 1} \to \Ker(\mathcal{F}^n \to \mathcal{F}^{n + 1})$ is surjective by the assumed vanishing of $H^n(\mathcal{F}^\bullet)$. If $\mathcal{E}^i$ is zero except for $i \in [a, b]$, then we have a split exact sequence of complexes $$ 0 \to \mathcal{E}^b[-b] \to \mathcal{E}^\bullet \to \sigma_{\leq b - 1}\mathcal{E}^\bullet \to 0 $$ which determines a distinguished triangle in $K(\mathcal{O}_U)$. Hence an exact sequence $$ \Hom_{K(\mathcal{O}_U)}( \sigma_{\leq b - 1}\mathcal{E}^\bullet, \mathcal{F}^\bullet) \to \Hom_{K(\mathcal{O}_U)}(\mathcal{E}^\bullet, \mathcal{F}^\bullet) \to \Hom_{K(\mathcal{O}_U)}(\mathcal{E}^b[-b], \mathcal{F}^\bullet) $$ by the axioms of triangulated categories. The composition $\mathcal{E}^b[-b] \to \mathcal{F}^\bullet$ is homotopic to zero on the members of a covering of $U$ by the above, whence we may assume our map comes from an element in the left hand side of the displayed exact sequence above. This element is zero on the members of a covering of $U$ by induction hypothesis.
Our assumptions on $f$ imply the cone $C(f)^\bullet$ has vanishing cohomology sheaves in degrees $\geq a$. Hence Lemma \ref{lemma-local-homotopy} guarantees there is a covering $\{U_i \to U\}$ such that the composition $\mathcal{E}^\bullet \to \mathcal{F}^\bullet \to C(f)^\bullet$ is homotopic to zero over $U_i$. Since $$ \mathcal{G}^\bullet \to \mathcal{F}^\bullet \to C(f)^\bullet \to \mathcal{G}^\bullet[1] $$ restricts to a distinguished triangle in $K(\mathcal{O}_{U_i})$ we see that we can lift $\alpha|_{U_i}$ up to homotopy to a map $\alpha_i : \mathcal{E}^\bullet|_{U_i} \to \mathcal{G}^\bullet|_{U_i}$ as desired.
Choose a quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{I}^\bullet$ into a K-injective complex. Let $(\mathcal{H}')^\bullet$ be the complex with terms $$ (\mathcal{H}')^n = \prod\nolimits_{n = p + q} \SheafHom_\mathcal{O}(\mathcal{L}^{-q}, \mathcal{I}^p) $$ which represents $R\SheafHom(\mathcal{E}^\bullet, \mathcal{F}^\bullet)$ by the construction in Section \ref{section-internal-hom}. It suffices to show that the map $$ \mathcal{H}^\bullet \longrightarrow (\mathcal{H}')^\bullet $$ is a quasi-isomorphism. Given an object $U$ of $\mathcal{C}$ we have by inspection $$ H^0(\mathcal{H}^\bullet(U)) = \Hom_{K(\mathcal{O}_U)}(\mathcal{E}^\bullet|_U, \mathcal{K}^\bullet|_U) \to H^0((\mathcal{H}')^\bullet(U)) = \Hom_{D(\mathcal{O}_U)}(\mathcal{E}^\bullet|_U, \mathcal{K}^\bullet|_U) $$ By Lemma \ref{lemma-local-actual} the sheafification of $U \mapsto H^0(\mathcal{H}^\bullet(U))$ is equal to the sheafification of $U \mapsto H^0((\mathcal{H}')^\bullet(U))$. A similar argument can be given for the other cohomology sheaves. Thus $\mathcal{H}^\bullet$ is quasi-isomorphic to $(\mathcal{H}')^\bullet$ which proves the lemma.
Choose a quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{I}^\bullet$ where $\mathcal{I}^\bullet$ is a bounded below complex of injectives. Note that $\mathcal{I}^\bullet$ is K-injective (Derived Categories, Lemma \ref{derived-lemma-bounded-below-injectives-K-injective}). Hence the construction in Section \ref{section-internal-hom} shows that $R\SheafHom(\mathcal{E}^\bullet, \mathcal{F}^\bullet)$ is  represented by the complex $(\mathcal{H}')^\bullet$ with terms $$ (\mathcal{H}')^n = \prod\nolimits_{n = p + q} \SheafHom_\mathcal{O}(\mathcal{E}^{-q}, \mathcal{I}^p) = \bigoplus\nolimits_{n = p + q} \SheafHom_\mathcal{O}(\mathcal{E}^{-q}, \mathcal{I}^p) $$ (equality because there are only finitely many nonzero terms). Note that $\mathcal{H}^\bullet$ is the total complex associated to the double complex with terms $\SheafHom_\mathcal{O}(\mathcal{E}^{-q}, \mathcal{F}^p)$ and similarly for $(\mathcal{H}')^\bullet$. The natural map $(\mathcal{H}')^\bullet \to \mathcal{H}^\bullet$ comes from a map of double complexes. Thus to show this map is a quasi-isomorphism, we may use the spectral sequence of a double complex (Homology, Lemma \ref{homology-lemma-first-quadrant-ss}) $$ {}'E_1^{p, q} = H^p(\SheafHom_\mathcal{O}(\mathcal{E}^{-q}, \mathcal{F}^\bullet)) $$ converging to $H^{p + q}(\mathcal{H}^\bullet)$ and similarly for $(\mathcal{H}')^\bullet$. To finish the proof of the lemma it suffices to show that $\mathcal{F}^\bullet \to \mathcal{I}^\bullet$ induces an isomorphism $$ H^p(\SheafHom_\mathcal{O}(\mathcal{E}, \mathcal{F}^\bullet)) \longrightarrow H^p(\SheafHom_\mathcal{O}(\mathcal{E}, \mathcal{I}^\bullet)) $$ on cohomology sheaves whenever $\mathcal{E}$ is a direct summand of a finite free $\mathcal{O}$-module. Since this is clear when $\mathcal{E}$ is finite free the result follows.
Let $\mathcal{F}^\bullet$ be any complex representing $E$ and let $X$, $\{U_i \to X\}$, and $\alpha_i : \mathcal{E}_i \to E|_{U_i}$ be as in (1). We will show that $\mathcal{F}^\bullet$ is $m$-pseudo-coherent as a complex, which will prove (1) and (2) in case $\mathcal{C}$ has a final object. By Lemma \ref{lemma-local-actual} we can after refining the covering $\{U_i \to X\}$ represent the maps $\alpha_i$ by maps of complexes $\alpha_i : \mathcal{E}_i^\bullet \to \mathcal{F}^\bullet|_{U_i}$. By assumption $H^j(\alpha_i)$ are isomorphisms for $j > m$, and $H^m(\alpha_i)$ is surjective whence $\mathcal{F}^\bullet$ is $m$-pseudo-coherent. \medskip\noindent Proof of (2). By the above we see that $\mathcal{F}^\bullet|_U$ is $m$-pseudo-coherent as a complex of $\mathcal{O}_U$-modules for all objects $U$ of $\mathcal{C}$. It is a formal consequence of the definitions that $\mathcal{F}^\bullet$ is $m$-pseudo-coherent. \medskip\noindent Proof of (3). Follows from the definitions and Sites, Definition \ref{sites-definition-site} part (2).
Say $f$ is given by the functor $u : \mathcal{D} \to \mathcal{C}$. Let $U$ be an object of $\mathcal{C}$. By Sites, Lemma \ref{sites-lemma-morphism-of-sites-covering} we can find a covering $\{U_i \to U\}$ and for each $i$ a morphism $U_i \to u(V_i)$ for some object $V_i$ of $\mathcal{D}$. By Lemma \ref{lemma-pseudo-coherent-independent-representative} it suffices to show that $Lf^*E|_{U_i}$ is $m$-pseudo-coherent. To do this it is enough to show that $Lf^*E|_{u(V_i)}$ is $m$-pseudo-coherent, since $Lf^*E|_{U_i}$ is the restriction of $Lf^*E|_{u(V_i)}$ to $\mathcal{C}/U_i$ (via Modules on Sites, Lemma \ref{sites-modules-lemma-relocalize}). By the commutative diagram of Modules on Sites, Lemma \ref{sites-modules-lemma-localize-morphism-ringed-sites} it suffices to prove the lemma for the morphism of ringed sites $(\mathcal{C}/u(V_i), \mathcal{O}_{u(V_i)}) \to (\mathcal{D}/V_i, \mathcal{O}_{V_i})$. Thus we may assume $\mathcal{D}$ has a final object $Y$ such that $X = u(Y)$ is a final object of $\mathcal{C}$. \medskip\noindent Let $\{V_i \to Y\}$ be a covering such that for each $i$ there exists a strictly perfect complex $\mathcal{F}_i^\bullet$ of $\mathcal{O}_{V_i}$-modules and a morphism $\alpha_i : \mathcal{F}_i^\bullet \to E|_{V_i}$ of $D(\mathcal{O}_{V_i})$ such that $H^j(\alpha_i)$ is an isomorphism for $j > m$ and $H^m(\alpha_i)$ is surjective. Arguing as above it suffices to prove the result for $(\mathcal{C}/u(V_i), \mathcal{O}_{u(V_i)}) \to (\mathcal{D}/V_i, \mathcal{O}_{V_i})$. Hence we may assume that there exists a strictly perfect complex $\mathcal{F}^\bullet$ of $\mathcal{O}_\mathcal{D}$-modules and a morphism $\alpha : \mathcal{F}^\bullet \to E$ of $D(\mathcal{O}_\mathcal{D})$ such that $H^j(\alpha)$ is an isomorphism for $j > m$ and $H^m(\alpha)$ is surjective. In this case, choose a distinguished triangle $$ \mathcal{F}^\bullet \to E \to C \to \mathcal{F}^\bullet[1] $$ The assumption on $\alpha$ means exactly that the cohomology sheaves $H^j(C)$ are zero for all $j \geq m$. Applying $Lf^*$ we obtain the distinguished triangle $$ Lf^*\mathcal{F}^\bullet \to Lf^*E \to Lf^*C \to Lf^*\mathcal{F}^\bullet[1] $$ By the construction of $Lf^*$ as a left derived functor we see that $H^j(Lf^*C) = 0$ for $j \geq m$ (by the dual of Derived Categories, Lemma \ref{derived-lemma-negative-vanishing}). Hence $H^j(Lf^*\alpha)$ is an isomorphism for $j > m$ and $H^m(Lf^*\alpha)$ is surjective. On the other hand, since $\mathcal{F}^\bullet$ is a bounded above complex of flat $\mathcal{O}_\mathcal{D}$-modules we see that $Lf^*\mathcal{F}^\bullet = f^*\mathcal{F}^\bullet$. Applying Lemma \ref{lemma-strictly-perfect-pullback} we conclude.