hcju/stacksqa huggingface dataset:
Part (1) follows from a combination of Lemmas \ref{lemma-check-lqc-on-etale-covering} and \ref{lemma-check-flat-comparison-on-etale-covering}. The proof of (2) is analogous to the proof of Lemma \ref{lemma-check-lqc-on-flat-covering}. Let $\mathcal{F}$ of a sheaf of $\mathcal{O}_\mathcal{X}$-modules on $\mathcal{X}_{fppf}$. \medskip\noindent First, suppose there is a morphism $a : \mathcal{U} \to \mathcal{X}$ which is surjective, flat, locally of finite presentation, quasi-compact, and quasi-separated such that $a^*\mathcal{F}$ is locally quasi-coherent and has the flat base change property. Then there is an exact sequence $$ 0 \to \mathcal{F} \to a_*a^*\mathcal{F} \to b_*b^*\mathcal{F} $$ where $b$ is the morphism $b : \mathcal{U} \times_\mathcal{X} \mathcal{U} \to \mathcal{X}$, see Sheaves on Stacks, Proposition \ref{stacks-sheaves-proposition-exactness-cech-complex} and Lemma \ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation}. Moreover, the pullback $b^*\mathcal{F}$ is the pullback of $a^*\mathcal{F}$ via one of the projection morphisms, hence is locally quasi-coherent and has the flat base change property, see Proposition \ref{proposition-lcq-flat-base-change}. The modules $a_*a^*\mathcal{F}$ and $b_*b^*\mathcal{F}$ are locally quasi-coherent and have the flat base change property by Proposition \ref{proposition-lcq-flat-base-change}. We conclude that $\mathcal{F}$ is locally quasi-coherent and has the flat base change property by Proposition \ref{proposition-lcq-flat-base-change}. \medskip\noindent Choose a scheme $U$ and a surjective smooth morphism $x : U \to \mathcal{X}$. By part (1) it suffices to show that $x^*\mathcal{F}$ is locally quasi-coherent and has the flat base change property. Again by part (1) it suffices to do this (Zariski) locally on $U$, hence we may assume that $U$ is affine. By Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-surjective-family-flat-locally-finite-presentation} there exists an fppf covering $\{a_i : U_i \to U\}$ such that each $x \circ a_i$ factors through some $f_j$. Hence the module $a_i^*\mathcal{F}$ on $(\Sch/U_i)_{fppf}$ is locally quasi-coherent and has the flat base change property. After refining the covering we may assume $\{U_i \to U\}_{i = 1, \ldots, n}$ is a standard fppf covering. Then $x^*\mathcal{F}$ is an fppf module on $(\Sch/U)_{fppf}$ whose pullback by the morphism $a : U_1 \amalg \ldots \amalg U_n \to U$ is locally quasi-coherent and has the flat base change property. Hence by the previous paragraph we see that $x^*\mathcal{F}$ is locally quasi-coherent and has the flat base change property as desired.
We first reduce (2) to (1). By Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-pushforward-restriction} we see that $R^if_*\mathcal{F}$ is the sheaf associated to the presheaf $$ y \longmapsto H^i_\tau\Big(V \times_{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{F}\Big) $$ Here $y$ is a typical object of $\mathcal{Y}$ lying over the scheme $V$. By Lemma \ref{lemma-parasitic} it suffices to show that these cohomology groups are zero when $y : V \to \mathcal{Y}$ is flat. Note that $\text{pr} : V \times_{y, \mathcal{Y}} \mathcal{X} \to \mathcal{X}$ is flat as a base change of $y$. Hence by Lemma \ref{lemma-parasitic} we see that $\text{pr}^{-1}\mathcal{F}$ is parasitic. Thus it suffices to prove (1). \medskip\noindent To see (1) we can use the spectral sequence of Sheaves on Stacks, Proposition \ref{stacks-sheaves-proposition-smooth-covering-compute-cohomology} to reduce this to the case where $\mathcal{X}$ is an algebraic stack representable by an algebraic space. Note that in the spectral sequence each $f_p^{-1}\mathcal{F} = f_p^*\mathcal{F}$ is a parasitic module by Lemma \ref{lemma-parasitic} because the morphisms $f_p : \mathcal{U}_p = \mathcal{U} \times_\mathcal{X} \ldots \times_\mathcal{X} \mathcal{U} \to \mathcal{X}$ are flat. Reusing this spectral sequence one more time (as in the proof of the key Lemma \ref{lemma-general-pushforward}) we reduce to the case where the algebraic stack $\mathcal{X}$ is representable by a scheme $X$. Then $H^i_\tau(\mathcal{X}, \mathcal{F}) = H^i((\Sch/X)_\tau, \mathcal{F})$. In this case the vanishing follows easily from an argument with {\v C}ech coverings, see Descent, Lemma \ref{descent-lemma-cohomology-parasitic}.
Let $x : U \to \mathcal{X}$ be a flat morphism where $U$ is a scheme. Then $x^*\mathcal{F}^\bullet$ is exact by Lemma \ref{lemma-flat-pullback-quasi-coherent}. Hence the restriction $x^*\mathcal{F}^\bullet|_{U_\etale}$ is exact which is what we had to prove.
Choose a scheme $U$ and a surjective smooth morphism $f : U \to \mathcal{X}$. Set $R = U \times_\mathcal{X} U$ so that we obtain a smooth groupoid $(U, R, s, t, c)$ in algebraic spaces with the property that $\mathcal{X} = [U/R]$, see Algebraic Stacks, Lemma \ref{algebraic-lemma-stack-presentation}. We may and do replace $\mathcal{X}$ by $[U/R]$. In the proof of Sheaves on Stacks, Proposition \ref{stacks-sheaves-proposition-quasi-coherent} we constructed a functor $$ q_1 : \QCoh(U, R, s, t, c) \longrightarrow \QCoh(\mathcal{O}_\mathcal{X}). $$ The construction of the inverse functor in the proof of Sheaves on Stacks, Proposition \ref{stacks-sheaves-proposition-quasi-coherent} works for objects of $\mathcal{M}_\mathcal{X}$ and induces a functor $$ q_2 : \mathcal{M}_\mathcal{X} \longrightarrow \QCoh(U, R, s, t, c). $$ Namely, if $\mathcal{F}$ is an object of $\mathcal{M}_\mathcal{X}$ the we set $$ q_2(\mathcal{F}) = (f^*\mathcal{F}|_{U_\etale}, \alpha) $$ where $\alpha$ is the isomorphism $$ t_{small}^*(f^*\mathcal{F}|_{U_\etale}) \to t^*f^*\mathcal{F}|_{R_\etale} \to s^*f^*\mathcal{F}|_{R_\etale} \to s_{small}^*(f^*\mathcal{F}|_{U_\etale}) $$ where the outer two morphisms are the comparison maps. Note that $q_2(\mathcal{F})$ is quasi-coherent precisely because $\mathcal{F}$ is locally quasi-coherent (and we used the flat base change property in the construction of the descent datum $\alpha$). We omit the verification that the cocycle condition (see Groupoids in Spaces, Definition \ref{spaces-groupoids-definition-groupoid-module}) holds. We define $Q = q_1 \circ q_2$. Let $\mathcal{F}$ be an object of $\mathcal{M}_\mathcal{X}$ and let $\mathcal{G}$ be an object of $\QCoh(\mathcal{O}_\mathcal{X})$. We have \begin{align*} \Mor_{\mathcal{M}_\mathcal{X}}(i(\mathcal{G}), \mathcal{F}) & = \Mor_{\QCoh(U, R, s, t, c)}(q_2(\mathcal{G}), q_2(\mathcal{F})) \\ & = \Mor_{\QCoh(\mathcal{O}_\mathcal{X})}(\mathcal{G}, Q(\mathcal{F})) \end{align*} where the first equality is Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-map-from-quasi-coherent} and the second equality holds because $q_1$ and $q_2$ are inverse equivalences of categories. The assertion $Q \circ i \cong \text{id}$ is a formal consequence of the fact that $i$ is fully faithful.
Write $\mathcal{X} = [U/R]$ as in the proof of Lemma \ref{lemma-adjoint}. Let $\mathcal{F}$ be an object of $\mathcal{M}_\mathcal{X}$. It is clear from the proof of Lemma \ref{lemma-adjoint} that $\mathcal{F}$ is in the kernel of $Q$ if and only if $\mathcal{F}|_{U_\etale} = 0$. In particular, if $\mathcal{F}$ is parasitic then $\mathcal{F}$ is in the kernel. Next, let $x : V \to \mathcal{X}$ be a flat morphism, where $V$ is a scheme. Set $W = V \times_\mathcal{X} U$ and consider the diagram $$ \xymatrix{ W \ar[d]_p \ar[r]_q & V \ar[d] \\ U \ar[r] & \mathcal{X} } $$ Note that the projection $p : W \to U$ is flat and the projection $q : W \to V$ is smooth and surjective. This implies that $q_{small}^*$ is a faithful functor on quasi-coherent modules. By assumption $\mathcal{F}$ has the flat base change property so that we obtain $p_{small}^*\mathcal{F}|_{U_\etale} \cong q_{small}^*\mathcal{F}|_{V_\etale}$. Thus if $\mathcal{F}$ is in the kernel of $Q$, then $\mathcal{F}|_{V_\etale} = 0$ which completes the proof of (1). \medskip\noindent Part (2) follows from the discussion above and the fact that the map $Q(\mathcal{F}) \to \mathcal{F}$ becomes an isomorphism after restricting to $U_\etale$. \medskip\noindent To see part (3) note that $Q$ is left exact as a right adjoint. Suppose that $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ is a short exact sequence in $\mathcal{M}_\mathcal{X}$. Let $\mathcal{E} = \Coker(Q(\mathcal{G}) \to Q(\mathcal{H}))$ in $\QCoh(\mathcal{O}_\mathcal{X})$. Since $\QCoh(\mathcal{O}_\mathcal{X}) \to \mathcal{M}_\mathcal{X}$ is a left adjoint it is right exact. Hence we see that $Q(\mathcal{G}) \to Q(\mathcal{H}) \to \mathcal{E} \to 0$ is exact in $\mathcal{M}_\mathcal{X}$. Using Lemma \ref{lemma-exact-sequence-quasi-coherent-parasitic-cohomology} we find that the top row of the following commutative diagram has parasitic cohomology sheaves at $Q(\mathcal{F})$ and $Q(\mathcal{G})$: $$ \xymatrix{ 0 \ar[r] & Q(\mathcal{F}) \ar[r] \ar[d]_a & Q(\mathcal{G}) \ar[r] \ar[d]_b & Q(\mathcal{H}) \ar[r] \ar[d]_c & \mathcal{E} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{F} \ar[r] & \mathcal{G} \ar[r] & \mathcal{H} \ar[r] & 0 } $$ The bottom row is exact and the vertical arrows $a, b, c$ have parasitic kernel and cokernels by part (2). It follows that $\mathcal{E}$ is parasitic: in the quotient category of $\textit{Mod}(\mathcal{O}_\mathcal{X})/\text{Parasitic}$ (see Homology, Lemma \ref{homology-lemma-serre-subcategory-is-kernel} and  Lemma \ref{lemma-parasitic}) we see that $a, b, c$ are isomorphisms and that the top row becomes exact. As it is also quasi-coherent, we conclude that $\mathcal{E}$ is zero because $\mathcal{E} = Q(\mathcal{E}) = 0$ by part (1).
By Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-Leray} the Leray spectral sequence with $$ E_2^{p, q} = H^p(\mathcal{Y}, R^qf_*\mathcal{F}) $$ converges to $H^{p + q}(\mathcal{X}, \mathcal{F})$. The kernel and cokernel of the adjunction map $$ R^qf_{\QCoh, *}\mathcal{F} \longrightarrow R^qf_*\mathcal{F} $$ are parasitic modules on $\mathcal{Y}$ (Lemma \ref{lemma-adjoint-kernel-parasitic}) hence have vanishing cohomology (Lemma \ref{lemma-pushforward-parasitic}). It follows formally that $H^p(\mathcal{Y}, R^qf_{\QCoh, *}\mathcal{F}) = H^p(\mathcal{Y}, R^qf_*\mathcal{F})$ and we win.
By Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-relative-Leray} the Leray spectral sequence with $$ E_2^{p, q} = R^pg_*(R^qf_*\mathcal{F}) $$ converges to $R^{p + q}(g \circ f)_*\mathcal{F}$. By the results of Proposition \ref{proposition-lcq-flat-base-change} all the terms of this spectral sequence are objects of $\mathcal{M}_\mathcal{Z}$. Applying the exact functor $Q_\mathcal{Z} : \mathcal{M}_\mathcal{Z} \to \QCoh(\mathcal{O}_\mathcal{Z})$ we obtain a spectral sequence in $\QCoh(\mathcal{O}_\mathcal{Z})$ covering to $R^{p + q}(g \circ f)_{\QCoh, *}\mathcal{F}$. Hence the result follows if we can show that $$ Q_\mathcal{Z}(R^pg_*(R^qf_*\mathcal{F})) = Q_\mathcal{Z}(R^pg_*(Q_\mathcal{X}(R^qf_*\mathcal{F})) $$ This follows from the fact that the kernel and cokernel of the map $$ Q_\mathcal{X}(R^qf_*\mathcal{F}) \longrightarrow R^qf_*\mathcal{F} $$ are parasitic (Lemma \ref{lemma-adjoint-kernel-parasitic}) and that $R^pg_*$ transforms parasitic modules into parasitic modules (Lemma \ref{lemma-pushforward-parasitic}).
In both cases it is immediate that the functor is fully faithful, continuous, and cocontinuous (see Sites, Definitions \ref{sites-definition-continuous} and \ref{sites-definition-cocontinuous}). Hence properties (a), (b), (c) follow from Sites, Lemmas \ref{sites-lemma-when-shriek} and \ref{sites-lemma-back-and-forth}. Parts (d), (e) follow from Modules on Sites, Lemmas \ref{sites-modules-lemma-g-shriek-adjoint} and \ref{sites-modules-lemma-back-and-forth}. Part (f) is immediate.
In both cases, the existence of the functor $g_!$ follows from Modules on Sites, Lemma \ref{sites-modules-lemma-lower-shriek-modules}. To see that $g_!$ agrees with the functor on abelian sheaves we will show the maps Modules on Sites, Equation (\ref{sites-modules-equation-compare-on-localizations}) are isomorphisms. \medskip\noindent Lisse-\'etale case. Let $x \in \Ob(\mathcal{X}_{lisse,\etale})$ lying over a scheme $U$ with $x : U \to \mathcal{X}$ smooth. Consider the induced fully faithful functor $$ g' : \mathcal{X}_{lisse,\etale}/x \longrightarrow \mathcal{X}_\etale/x $$ The right hand side is identified with $(\Sch/U)_\etale$ and the left hand side with the full subcategory of schemes $U'/U$ such that the composition $U' \to U \to \mathcal{X}$ is smooth. Thus \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-structure-sheaves} applies. \medskip\noindent Flat-fppf case. Let $x \in \Ob(\mathcal{X}_{flat,fppf})$ lying over a scheme $U$ with $x : U \to \mathcal{X}$ flat. Consider the induced fully faithful functor $$ g' : \mathcal{X}_{flat,fppf}/x \longrightarrow \mathcal{X}_{fppf}/x $$ The right hand side is identified with $(\Sch/U)_{fppf}$ and the left hand side with the full subcategory of schemes $U'/U$ such that the composition $U' \to U \to \mathcal{X}$ is flat. Thus \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-structure-sheaves} applies. \medskip\noindent In both cases the equality $g^*g_! = \text{id}$ follows from $g^* = g^{-1}$ and the equality for abelian sheaves in Lemma \ref{lemma-lisse-etale}.
Proof of (1). We may replace each of the algebraic stacks $\mathcal{X}_j$ by a scheme $U_j$ (using that any algebraic stack has a smooth covering by a scheme and that compositions of smooth morphisms are smooth, see Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-composition-smooth}). The pullback of $\mathcal{F}$ to $(\Sch/U_j)_\etale$ is still quasi-coherent, see Modules on Sites, Lemma \ref{sites-modules-lemma-local-pullback}. Then $f = \coprod f_j : U = \coprod U_j \to \mathcal{X}$ is a smooth surjective morphism. Let $x : V \to \mathcal{X}$ be an object of $\mathcal{X}$. By Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation} there exists an \'etale covering $\{x_i \to x\}_{i \in I}$ such that each $x_i$ lifts to an object $u_i$ of $(\Sch/U)_\etale$. This just means that $x_i$ lives over a scheme $V_i$, that $\{V_i \to V\}$ is an \'etale covering, and that $x_i$ comes from a morphism $u_i : V_i \to U$. Then $x_i^*\mathcal{F} = u_i^*f^*\mathcal{F}$ is quasi-coherent. This implies that $x^*\mathcal{F}$ on $(\Sch/V)_\etale$ is quasi-coherent, for example by Modules on Sites, Lemma \ref{sites-modules-lemma-local-final-object}. By Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-characterize-quasi-coherent-bis} we see that $x^*\mathcal{F}$ is an fppf sheaf and since $x$ was arbitrary we see that $\mathcal{F}$ is a sheaf in the fppf topology. Applying Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-characterize-quasi-coherent} we see that $\mathcal{F}$ is quasi-coherent. \medskip\noindent Proof of (2). This is proved using exactly the same argument, which we fully write out here. We may replace each of the algebraic stacks $\mathcal{X}_j$ by a scheme $U_j$ (using that any algebraic stack has a smooth covering by a scheme and that flat and locally finite presented morphisms are preserved by composition, see Morphisms of Stacks, Lemmas \ref{stacks-morphisms-lemma-composition-flat} and \ref{stacks-morphisms-lemma-composition-finite-presentation}). The pullback of $\mathcal{F}$ to $(\Sch/U_j)_\etale$ is still locally quasi-coherent, see Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-pullback-quasi-coherent}. Then $f = \coprod f_j : U = \coprod U_j \to \mathcal{X}$ is a surjective, flat, and locally finitely presented morphism. Let $x : V \to \mathcal{X}$ be an object of $\mathcal{X}$. By Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation} there exists an fppf covering $\{x_i \to x\}_{i \in I}$ such that each $x_i$ lifts to an object $u_i$ of $(\Sch/U)_\etale$. This just means that $x_i$ lives over a scheme $V_i$, that $\{V_i \to V\}$ is an fppf covering, and that $x_i$ comes from a morphism $u_i : V_i \to U$. Then $x_i^*\mathcal{F} = u_i^*f^*\mathcal{F}$ is quasi-coherent. This implies that $x^*\mathcal{F}$ on $(\Sch/V)_\etale$ is quasi-coherent, for example by Modules on Sites, Lemma \ref{sites-modules-lemma-local-final-object}. By Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-characterize-quasi-coherent} we see that $\mathcal{F}$ is quasi-coherent.
Pick a scheme $U$ and a surjective smooth morphism $x : U \to \mathcal{X}$. By Modules on Sites, Definition \ref{sites-modules-definition-site-local} there exists an \'etale (resp.\ fppf) covering $\{U_i \to U\}_{i \in I}$ such that each pullback $f_i^{-1}\mathcal{H}$ has a global presentation (see Modules on Sites, Definition \ref{sites-modules-definition-global}). Here $f_i : U_i \to \mathcal{X}$ is the composition $U_i \to U \to \mathcal{X}$ which is a morphism of algebraic stacks. (Recall that the pullback ``is'' the restriction to $\mathcal{X}/f_i$, see Sheaves on Stacks, Definition \ref{stacks-sheaves-definition-pullback} and the discussion following.) Since each $f_i$ is smooth (resp.\ flat) by Lemma \ref{lemma-lisse-etale-functorial} we see that $f_i^{-1}g_!\mathcal{H} = g_{i, !}(f'_i)^{-1}\mathcal{H}$. Using Lemma \ref{lemma-check-qc-on-etale-covering} we reduce the statement of the lemma to the case where $\mathcal{H}$ has a global presentation. Say we have $$ \bigoplus\nolimits_{j \in J} \mathcal{O} \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow \mathcal{H} \longrightarrow 0 $$ of $\mathcal{O}$-modules where $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{lisse,\etale}}$ (resp.\ $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}$). Since $g_!$ commutes with arbitrary colimits (as a left adjoint functor, see Lemma \ref{lemma-lisse-etale-modules} and Categories, Lemma \ref{categories-lemma-adjoint-exact}) we conclude that there exists an exact sequence $$ \bigoplus\nolimits_{j \in J} g_!\mathcal{O} \longrightarrow \bigoplus\nolimits_{i \in I} g_!\mathcal{O} \longrightarrow g_!\mathcal{H} \longrightarrow 0 $$ Lemma \ref{lemma-lisse-etale-structure-sheaf} shows that $g_!\mathcal{O} = \mathcal{O}_\mathcal{X}$. In case (2) we are done. In case (1) we apply Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-characterize-quasi-coherent-bis} to conclude.
Pullback by any morphism of ringed topoi preserves categories of quasi-coherent modules, see Modules on Sites, Lemma \ref{sites-modules-lemma-local-pullback}. Hence $g^{-1}$ preserves the categories of quasi-coherent modules; here we use that $\QCoh(\mathcal{O}_\mathcal{X}) = \QCoh(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ by Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-characterize-quasi-coherent-bis}. The same is true for $g_!$ by Lemma \ref{lemma-shriek-quasi-coherent}. We know that $\mathcal{H} \to g^{-1}g_!\mathcal{H}$ is an isomorphism by Lemma \ref{lemma-lisse-etale}. Conversely, if $\mathcal{F}$ is in $\QCoh(\mathcal{O}_\mathcal{X})$ then the map $g_!g^{-1}\mathcal{F} \to \mathcal{F}$ is a map of quasi-coherent modules on $\mathcal{X}$ whose restriction to any scheme smooth over $\mathcal{X}$ is an isomorphism. Then the discussion in Sheaves on Stacks, Sections \ref{stacks-sheaves-section-quasi-coherent-presentation} and \ref{stacks-sheaves-section-quasi-coherent-algebraic-stacks} (comparing with quasi-coherent modules on presentations) shows it is an isomorphism. This proves (1)(a) and (2)(a). \medskip\noindent Let $\mathcal{F}$ be an object of $\mathcal{M}_\mathcal{X}$. By Lemma \ref{lemma-adjoint-kernel-parasitic} the kernel and cokernel of the map $Q(\mathcal{F}) \to \mathcal{F}$ are parasitic. Hence by Lemma \ref{lemma-parasitic-in-terms-flat-fppf} and since $g^* = g^{-1}$ is exact, we conclude $g^*Q(\mathcal{F}) \to g^*\mathcal{F}$ is an isomorphism. Thus $g^*\mathcal{F}$ is quasi-coherent. This proves (1)(b) and (2)(b). Finally, (1)(c) and (2)(c) follow because $g_!g^*Q(\mathcal{F}) \to Q(\mathcal{F})$ is an isomorphism by our arguments above.
Part (1) is Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-lqc-flat-base-change-fppf-sheaf}. \medskip\noindent Part (2) for the embedding $\mathcal{M}_\mathcal{X} \subset \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ we have seen in the proof of Lemma \ref{lemma-flat-comparison}. Let us prove (2) for the embedding $\mathcal{M}_\mathcal{X} \subset \textit{Mod}(\mathcal{O}_\mathcal{X})$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism between objects of $\mathcal{M}_\mathcal{X}$. Since $\Ker(\varphi)$ is the same whether computed in the \'etale or the fppf topology, we see that $\Ker(\varphi)$ is in $\mathcal{M}_\mathcal{X}$ by the \'etale case. On the other hand, the cokernel computed in the fppf topology is the fppf sheafification of the cokernel computed in the \'etale topology. However, this \'etale cokernel is in $\mathcal{M}_\mathcal{X}$ hence an fppf sheaf by (1) and we see that the cokernel is in $\mathcal{M}_\mathcal{X}$. Finally, suppose that $$ 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 $$ is an exact sequence in $\textit{Mod}(\mathcal{O}_\mathcal{X})$ (i.e., using the fppf topology) with $\mathcal{F}_1$, $\mathcal{F}_2$ in $\mathcal{M}_\mathcal{X}$. In order to show that $\mathcal{F}_2$ is an object of $\mathcal{M}_\mathcal{X}$ it suffices to show that the sequence is also exact in the \'etale topology. To do this it suffices to show that any element of $H^1_{fppf}(x, \mathcal{F}_1)$ becomes zero on the members of an \'etale covering of $x$ (for any object $x$ of $\mathcal{X}$). This is true because $H^1_{fppf}(x, \mathcal{F}_1) = H^1_\etale(x, \mathcal{F}_1)$ by Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-compare-fppf-etale} and because of locality of cohomology, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-kill-cohomology-class-on-covering}. This proves (2). \medskip\noindent Part (3) follows from Lemma \ref{lemma-check-flat-comparison-on-etale-covering} and Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-pullback-lqc}. \medskip\noindent Part (4)(b) for $R^if_*\mathcal{F}$ computed in the \'etale cohomology follows from Lemma \ref{lemma-flat-comparison}. Whereupon part (4)(a) follows from Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-compare-fppf-etale} combined with (1) above. \medskip\noindent Part (5) for the \'etale topology follows from Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-lqc-colimits} and Lemma \ref{lemma-check-flat-comparison-on-etale-covering}. The fppf version then follows as the colimit in the \'etale topology is already an fppf sheaf by part (1).
Note that each of the morphisms $f_p : \mathcal{U} \times_\mathcal{X} \ldots \times_\mathcal{X} \mathcal{U} \to \mathcal{X}$ is quasi-compact and quasi-separated, hence $g \circ f_p$ is quasi-compact and quasi-separated, hence the assertion makes sense (i.e., the functors $R^q(g \circ f_p)_{\QCoh, *}$ are defined). There is a spectral sequence $$ E_2^{p, q} = R^q(g \circ f_p)_*f_p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F} $$ by Sheaves on Stacks, Proposition \ref{stacks-sheaves-proposition-smooth-covering-compute-direct-image}. Applying the exact functor $Q_\mathcal{Y} : \mathcal{M}_\mathcal{Y} \to \QCoh(\mathcal{O}_\mathcal{Y})$ gives the desired spectral sequence in $\QCoh(\mathcal{O}_\mathcal{Y})$.