hcju/stacksqa huggingface dataset:
Part (1) follows Categories, Lemma \ref{categories-lemma-fibred-groupoids-fibre-product-goes-up} as $(\Sch/S)_{fppf}$ has fibre products. \medskip\noindent Let $a, b : x \to y$ be morphisms of $\mathcal{X}$. Set $U = p(x)$ and $V = p(y)$. The category of schemes has equalizers hence we can let $W \to U$ be the equalizer of $p(a)$ and $p(b)$. Denote $c : z \to x$ a morphism of $\mathcal{X}$ lying over $W \to U$. The equalizer of $a$ and $b$, if it exists, is the equalizer of $a \circ c$ and $b \circ c$. Thus we may assume that $p(a) = p(b) = f : U \to V$. As $\mathcal{X}$ is fibred in groupoids, there exists a unique automorphism $i : x \to x$ in the fibre category of $\mathcal{X}$ over $U$ such that $a \circ i = b$. Again the equalizer of $a$ and $b$ is the equalizer of $\text{id}_x$ and $i$. Recall that the $\mathit{Isom}_\mathcal{X}(x)$ is the presheaf on $(\Sch/U)_{fppf}$ which to $T/U$ associates the set of automorphisms of $x|_T$ in the fibre category of $\mathcal{X}$ over $T$, see Stacks, Definition \ref{stacks-definition-mor-presheaf}. If $\mathit{Isom}_\mathcal{X}(x)$ is representable by an algebraic space $G \to U$, then we see that $\text{id}_x$ and $i$ define morphisms $e, i : U \to G$ over $U$. Set $M = U \times_{e, G, i} U$, which by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-section-immersion} is a scheme. Then it is clear that $x|_M \to x$ is the equalizer of the maps $\text{id}_x$ and $i$ in $\mathcal{X}$. This proves (2). \medskip\noindent If $\mathcal{X} = [U/R]$ for some groupoid in algebraic spaces $(U, R, s, t, c)$ over $S$, then the hypothesis of (2) holds by Bootstrap, Lemma \ref{bootstrap-lemma-quotient-stack-isom}. If $\mathcal{X}$ is an algebraic stack, then we can choose a presentation $[U/R] \cong \mathcal{X}$ by Algebraic Stacks, Lemma \ref{algebraic-lemma-stack-presentation}.
By Categories, Lemma \ref{categories-lemma-fibred-groupoids-fibre-product-goes-up} we see that a fibre product in $\mathcal{X}$ is any commutative square lying over a fibre product diagram in $(\Sch/S)_{fppf}$. Similarly for $\mathcal{Y}$. Hence (1) is clear. \medskip\noindent Let $x \to x'$ be the equalizer of two morphisms $a, b : x' \to x''$ in $\mathcal{X}$. We will show that $f(x) \to f(x')$ is the equalizer of $f(a)$ and $f(b)$. Let $y \to f(x)$ be a morphism of $\mathcal{Y}$ equalizing $f(a)$ and $f(b)$. Say $x, x', x''$ lie over the schemes $U, U', U''$ and $y$ lies over $V$. Denote $h : V \to U'$ the image of $y \to f(x)$ in the category of schemes. The morphism $y \to f(x)$ is isomorphic to $f(h^*x') \to f(x')$ by the axioms of fibred categories. Hence, as $f$ is faithful, we see that $h^*x' \to x'$ equalizes $a$ and $b$. Thus we obtain a unique morphism $h^*x' \to x$ whose image $y = f(h^*x') \to f(x)$ is the desired morphism in $\mathcal{Y}$.
We think of objects in $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$ as quadruples $(U, x, z, \alpha)$ where $\alpha : f(x) \to g(z)$ is an isomorphism over $U$, see Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}. A morphism $(U, x, z, \alpha) \to (U', x', z', \alpha')$ is a pair of morphisms $a : x \to x'$ and $b : z \to z'$ compatible with $\alpha$ and $\alpha'$. Thus it is clear that if $f$ and $g$ are faithful, so is the functor $\mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Y}$. Now, suppose that $(a, b), (a', b') : (U, x, z, \alpha) \to (U', x', z', \alpha')$ are two morphisms of the $2$-fibre product. Then consider the equalizer $x'' \to x$ of $a$ and $a'$ and the equalizer $z'' \to z$ of $b$ and $b'$. Since $f$ commutes with equalizers (by Lemma \ref{lemma-fibre-products-morphism}) we see that $f(x'') \to f(x)$ is the equalizer of $f(a)$ and $f(a')$. Similarly, $g(z'') \to g(z)$ is the equalizer of $g(b)$ and $g(b')$. Picture $$ \xymatrix{ f(x'') \ar[r] \ar@{..>}[d]_{\alpha''}& f(x) \ar[d]_\alpha \ar@<0.5ex>[r]^{f(a)} \ar@<-0.5ex>[r]_{f(a')}  & f(x') \ar[d]^{\alpha'} \\ g(z'') \ar[r] & g(z) \ar@<0.5ex>[r]^{g(b)} \ar@<-0.5ex>[r]_{g(b')}  & g(z') } $$ It is clear that the dotted arrow exists and is an isomorphism. However, it is not a priori the case that the image of $\alpha''$ in the category of schemes is the identity of its source. On the other hand, the existence of $\alpha''$ means that we can assume that $x''$ and $z''$ are defined over the same scheme and that the morphisms $x'' \to x$ and $z'' \to z$ have the same image in the category of schemes. Redoing the diagram above we see that the dotted arrow now does project to an identity morphism and we win. Some details omitted.
By Stacks, Lemma \ref{stacks-lemma-topology-inherited-functorial} the functor $f$ is continuous and cocontinuous. Hence by Modules on Sites, Lemma \ref{sites-modules-lemma-g-shriek-adjoint} the functor $f^{-1} : \textit{Ab}(\mathcal{Y}_\tau) \to \textit{Ab}(\mathcal{X}_\tau)$ has a left adjoint $f_! : \textit{Ab}(\mathcal{X}_\tau) \to \textit{Ab}(\mathcal{Y}_\tau)$. To see (1) we apply Modules on Sites, Lemma \ref{sites-modules-lemma-exactness-lower-shriek} and to see that the hypotheses of that lemma are satisfied use Lemmas \ref{lemma-fibre-products} and \ref{lemma-fibre-products-morphism} above. Part (2) follows from this formally, see Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}.
Recall that $f$ is a continuous and cocontinuous functor of sites and that $f^{-1}\mathcal{O}_\mathcal{Y} = \mathcal{O}_\mathcal{X}$. Hence Modules on Sites, Lemma \ref{sites-modules-lemma-lower-shriek-modules} implies $f^*$ has a left adjoint $f_!^{Mod}$. Let $x$ be an object of $\mathcal{X}$ lying over the scheme $U$. Then $f$ induces an equivalence of ringed sites $$ \mathcal{X}/x \longrightarrow \mathcal{Y}/f(x) $$ as both sides are equivalent to $(\Sch/U)_\tau$, see Lemma \ref{lemma-localizing-structure-sheaf}. Modules on Sites, Remark \ref{sites-modules-remark-when-shriek-equal} shows that $f_!$ agrees with the functor on abelian sheaves. \medskip\noindent Assume now that $\mathcal{X}$ has equalizers and that $f$ is faithful. Lemma \ref{lemma-pullback-injective} tells us that $f_!$ is exact. Finally, Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives} implies the statement on pullbacks of injective modules.
Set $\mathcal{U}' = \mathcal{U} \times_\mathcal{X} \mathcal{X}$ equal to the fibre product as described in Categories, Lemma \ref{categories-lemma-2-product-categories-over-C}. Set $f' : \mathcal{U}' \to \mathcal{X}$ equal to the second projection. Then $\mathcal{U} \to \mathcal{U}'$, $u \mapsto (u, f(x), 1)$ is an equivalence over $\mathcal{X}$, hence we may replace $(\mathcal{U}, f)$ by $(\mathcal{U}', f')$ by Lemma \ref{lemma-generalities}. The advantage of this is that now $f'$ has a section $s'$ such that $f' \circ s' = \text{id}_\mathcal{X}$ on the nose. Namely, if $t : s \circ f \to \text{id}_\mathcal{X}$ is a $2$-isomorphism then we can set $s'(x) = (s(x), x, t_x)$. Thus we may assume that $f \circ s = \text{id}_\mathcal{X}$. \medskip\noindent In the case that $f \circ s = \text{id}_\mathcal{X}$ the result follows from general principles. We give the homotopy explicitly. Namely, for $n \geq 0$ define $s_n : \mathcal{U}_n \to \mathcal{U}_{n + 1}$ to be the $1$-morphism defined by the rule on objects $$ (u_0, \ldots, u_n, x, \alpha_0, \ldots, \alpha_n) \longmapsto (u_0, \ldots, u_n, s(x), x, \alpha_0, \ldots, \alpha_n, \text{id}_x). $$ Define $$ h^{n + 1} : \Gamma(\mathcal{U}_{n + 1}, f_{n + 1}^{-1}\mathcal{F}) \longrightarrow \Gamma(\mathcal{U}_n, f_n^{-1}\mathcal{F}) $$ as pullback along $s_n$. We also set $s_{-1} = s$ and $h^0 : \Gamma(\mathcal{U}_0, f_0^{-1}\mathcal{F}) \to \Gamma(\mathcal{X}, \mathcal{F})$ equal to pullback along $s_{-1}$. Then the family of maps $\{h^n\}_{n \geq 0}$ is a homotopy between $1$ and $0$ on the extended {\v C}ech complex.
Let $y$ be an object of $\mathcal{Y}$ lying over the scheme $T$. Set $x = e(y)$. We are going to show that the map induces an isomorphism on sections over $y$. Note that $$ \Gamma(y, e^{-1}\mathcal{K}^\bullet(f, \mathcal{F})) = \Gamma(x, \mathcal{K}^\bullet(f, \mathcal{F})) = \check{\mathcal{C}}^\bullet( (\Sch/T)_{fppf} \times_{x, \mathcal{X}} \mathcal{U} \to (\Sch/T)_{fppf}, x^{-1}\mathcal{F}) $$ by Remark \ref{remark-cech-complex-sections}. On the other hand, $$ \Gamma(y, \mathcal{K}^\bullet(g, e^{-1}\mathcal{F})) = \check{\mathcal{C}}^\bullet( (\Sch/T)_{fppf} \times_{y, \mathcal{Y}} \mathcal{V} \to (\Sch/T)_{fppf}, y^{-1}e^{-1}\mathcal{F}) $$ also by Remark \ref{remark-cech-complex-sections}. Note that $y^{-1}e^{-1}\mathcal{F} = x^{-1}\mathcal{F}$ and since the diagram is $2$-cartesian the $1$-morphism $$ (\Sch/T)_{fppf} \times_{y, \mathcal{Y}} \mathcal{V} \to (\Sch/T)_{fppf} \times_{x, \mathcal{X}} \mathcal{U} $$ is an equivalence. Hence the map on sections over $y$ is an isomorphism by Lemma \ref{lemma-generalities}.
Before we start the proof we make some remarks. By Lemma \ref{lemma-fibre-products-preserve-properties} (and induction) all of the categories fibred in groupoids $\mathcal{U}_p$ have equalizers and all of the morphisms $f_p : \mathcal{U}_p \to \mathcal{X}$ are faithful. Let $\mathcal{I}$ be an injective object of $\textit{Ab}(\mathcal{X}_\tau)$. By Lemma \ref{lemma-pullback-injective} we see $f_p^{-1}\mathcal{I}$ is an injective object of $\textit{Ab}((\mathcal{U}_p)_\tau)$. Hence $f_{p, *}f_p^{-1}\mathcal{I}$ is an injective object of $\textit{Ab}(\mathcal{X}_\tau)$ by Lemma \ref{lemma-pushforward-injective}. Hence Proposition \ref{proposition-exactness-cech-complex} shows that the extended relative {\v C}ech complex $$ \ldots \to 0 \to \mathcal{I} \to f_{0, *}f_0^{-1}\mathcal{I} \to f_{1, *}f_1^{-1}\mathcal{I} \to f_{2, *}f_2^{-1}\mathcal{I} \to \ldots $$ is an exact complex in $\textit{Ab}(\mathcal{X}_\tau)$ all of whose terms are injective. Taking global sections of this complex is exact and we see that the {\v C}ech complex $\check{\mathcal{C}}^\bullet(\mathcal{U} \to \mathcal{X}, \mathcal{I})$ is quasi-isomorphic to $\Gamma(\mathcal{X}_\tau, \mathcal{I})[0]$. \medskip\noindent With these preliminaries out of the way consider the two spectral sequences associated to the double complex (see Homology, Section \ref{homology-section-double-complex}) $$ \check{\mathcal{C}}^\bullet(\mathcal{U} \to \mathcal{X}, \mathcal{I}^\bullet) $$ where $\mathcal{F} \to \mathcal{I}^\bullet$ is an injective resolution in $\textit{Ab}(\mathcal{X}_\tau)$. The discussion above shows that Homology, Lemma \ref{homology-lemma-double-complex-gives-resolution} applies which shows that $\Gamma(\mathcal{X}_\tau, \mathcal{I}^\bullet)$ is quasi-isomorphic to the total complex associated to the double complex. By our remarks above the complex $f_p^{-1}\mathcal{I}^\bullet$ is an injective resolution of $f_p^{-1}\mathcal{F}$. Hence the other spectral sequence is as indicated in the lemma.
The proof of this lemma is identical to the proof of Lemma \ref{lemma-cech-to-cohomology} except that it uses an injective resolution in $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$ and it uses Lemma \ref{lemma-pullback-injective-modules} instead of Lemma \ref{lemma-pullback-injective}.
Proof of (1). Suppose that $y$ lies over the scheme $V$. We may think of $y$ as a morphism $(\Sch/V)_{fppf} \to \mathcal{Y}$. By definition the $2$-fibre product $\mathcal{X} \times_\mathcal{Y} (\Sch/V)_{fppf}$ is representable by an algebraic space $W$ and the morphism $W \to V$ is surjective, flat, and locally of finite presentation. Choose a scheme $U$ and a surjective \'etale morphism $U \to W$. Then $U \to V$ is also surjective, flat, and locally of finite presentation (see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-etale-flat}, \ref{spaces-morphisms-lemma-etale-locally-finite-presentation}, \ref{spaces-morphisms-lemma-composition-surjective}, \ref{spaces-morphisms-lemma-composition-finite-presentation}, and \ref{spaces-morphisms-lemma-composition-flat}). Hence $\{U \to V\}$ is an fppf covering. Denote $x$ the object of $\mathcal{X}$ over $U$ corresponding to the $1$-morphism $(\Sch/U)_{fppf} \to \mathcal{X}$. Then $\{f(x) \to y\}$ is the desired fppf covering of $\mathcal{Y}$. \medskip\noindent Proof of (2). Suppose that $y$ lies over the scheme $V$. We may think of $y$ as a morphism $(\Sch/V)_{fppf} \to \mathcal{Y}$. By definition the $2$-fibre product $\mathcal{X} \times_\mathcal{Y} (\Sch/V)_{fppf}$ is representable by an algebraic space $W$ and the morphism $W \to V$ is surjective and smooth. Choose a scheme $U$ and a surjective \'etale morphism $U \to W$. Then $U \to V$ is also surjective and smooth (see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-etale-smooth}, \ref{spaces-morphisms-lemma-composition-surjective}, and \ref{spaces-morphisms-lemma-composition-smooth}). Hence $\{U \to V\}$ is a smooth covering. By More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-dominates-smooth} there exists an \'etale covering $\{V_i \to V\}$ such that each $V_i \to V$ factors through $U$. Denote $x_i$ the object of $\mathcal{X}$ over $V_i$ corresponding to the $1$-morphism $$ (\Sch/V_i)_{fppf} \to (\Sch/U)_{fppf} \to \mathcal{X}. $$ Then $\{f(x_i) \to y\}$ is the desired \'etale covering of $\mathcal{Y}$.
Note that the assumptions on $f : \mathcal{U} \to \mathcal{X}$ and $\mathcal{F}$ are identical to those in Lemma \ref{lemma-cech-to-cohomology}. Hence the preliminary remarks made in the proof of that lemma hold here also. These remarks imply in particular that $$ 0 \to g_*\mathcal{I} \to (g \circ f_0)_*f_0^{-1}\mathcal{I} \to (g \circ f_1)_*f_1^{-1}\mathcal{I} \to \ldots $$ is exact if $\mathcal{I}$ is an injective object of $\textit{Ab}(\mathcal{X}_\tau)$. Having said this, consider the two spectral sequences of Homology, Section \ref{homology-section-double-complex} associated to the double complex $\mathcal{C}^{\bullet, \bullet}$ with terms $$ \mathcal{C}^{p, q} = (g \circ f_p)_*\mathcal{I}^q $$ where $\mathcal{F} \to \mathcal{I}^\bullet$ is an injective resolution in $\textit{Ab}(\mathcal{X}_\tau)$. The first spectral sequence implies, via Homology, Lemma \ref{homology-lemma-double-complex-gives-resolution}, that $g_*\mathcal{I}^\bullet$ is quasi-isomorphic to the total complex associated to $\mathcal{C}^{\bullet, \bullet}$. Since $f_p^{-1}\mathcal{I}^\bullet$ is an injective resolution of $f_p^{-1}\mathcal{F}$ (see Lemma \ref{lemma-pullback-injective}) the second spectral sequence has terms $E_1^{p, q} = R^q(g \circ f_p)_*f_p^{-1}\mathcal{F}$ as in the statement of the lemma.
The proof is identical to the proof of Lemma \ref{lemma-cech-to-cohomology-relative} except that it uses an injective resolution in $\textit{Mod}(\mathcal{X}_\tau, \mathcal{O}_\mathcal{X})$ and it uses Lemma \ref{lemma-pullback-injective-modules} instead of Lemma \ref{lemma-pullback-injective}.
Choose an injective resolution $\mathcal{F}[0] \to \mathcal{I}^\bullet$. By the formula for pushforward (\ref{equation-pushforward}) we see that $R^if_*\mathcal{F}$ is the sheaf associated to the presheaf which associates to $y$ the cohomology of the complex $$ \begin{matrix} \Gamma\Big((\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{I}^{i - 1}\Big) \\ \downarrow \\ \Gamma\Big((\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{I}^i\Big) \\ \downarrow \\ \Gamma\Big((\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{I}^{i + 1}\Big) \end{matrix} $$ Since $\text{pr}^{-1}$ is exact, it suffices to show that $\text{pr}^{-1}$ preserves injectives. This follows from Lemmas \ref{lemma-pullback-injective} and \ref{lemma-pullback-injective-modules} as well as the fact that $\text{pr}$ is a representable morphism of algebraic stacks (so that $\text{pr}$ is faithful by Algebraic Stacks, Lemma \ref{algebraic-lemma-characterize-representable-by-algebraic-spaces} and that $(\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}$ has equalizers by  Lemma \ref{lemma-fibre-products}).
The isomorphism $g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}$ is Lemma \ref{lemma-base-change} (and it holds for arbitrary presheaves). For the derived direct images, there is a base change map because the morphisms $g$ and $g'$ are flat, see Cohomology on Sites, Section \ref{sites-cohomology-section-base-change-map}. To see that this map is a quasi-isomorphism we can use that for an object $y'$ of $\mathcal{Y}'$ over a scheme $V$ there is an equivalence $$ (\Sch/V)_{fppf} \times_{g(y'), \mathcal{Y}} \mathcal{X} = (\Sch/V)_{fppf} \times_{y', \mathcal{Y}'} (\mathcal{Y}' \times_\mathcal{Y} \mathcal{X}) $$ We conclude that the induced map $g^{-1}R^if_*\mathcal{F} \to R^if'_*(g')^{-1}\mathcal{F}$ is an isomorphism by Lemma \ref{lemma-pushforward-restriction}.
This follows formally from the fact that the restriction functor $\pi_{F, *} = i_F^{-1}$ (see Lemma \ref{lemma-compare}) is right adjoint to the exact functor $\pi_F^{-1}$, see Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives} and Derived Categories, Lemma \ref{derived-lemma-adjoint-preserve-K-injectives}. To see that the lemma does not hold for modules, we refer the reader to \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-injectives}.
Part (1) follows immediately from Lemma \ref{lemma-compare-injectives} and (\ref{equation-compare-big-small}) on choosing an injective resolution of $\mathcal{F}$. \medskip\noindent Part (2) can be proved as follows. In Lemma \ref{lemma-compare-morphism} we have seen that $\pi_G \circ f = f_{small} \circ \pi_F$ as morphisms of ringed sites. Hence we obtain $R\pi_{G, *} \circ Rf_* = Rf_{small, *} \circ R\pi_{F, *}$ by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-derived-pushforward-composition}. Since the restriction functors $\pi_{F, *}$ and $\pi_{G, *}$ are exact, we conclude.
Let $\{x_i \to x\}$ be an fppf covering of $\mathcal{X}$ lying over the fppf covering $\{f_i : U_i \to U\}$ of schemes over $S$. By assumption the restriction $\mathcal{G} = \mathcal{F}|_{U_\etale}$ is quasi-coherent and the comparison maps $f_{i, small}^*\mathcal{G} \to \mathcal{F}|_{U_{i, \etale}}$ are isomorphisms. Hence the sheaf condition for $\mathcal{F}$ and the covering $\{x_i \to x\}$ is equivalent to the sheaf condition for $\mathcal{G}^a$ on $(\Sch/U)_{fppf}$ and the covering $\{U_i \to U\}$ which holds by Descent, Lemma \ref{descent-lemma-sheaf-condition-holds}.
The assertion that $\mathcal{F}$ is an $\mathcal{O}_\mathcal{X}$-module follows from Lemma \ref{lemma-lqc-flat-base-change-fppf-sheaf}. Note that $\epsilon$ is a morphism of sites given by the identity functor on $\mathcal{X}$. The sheaf $R^p\epsilon_*\mathcal{F}$ is therefore the sheaf associated to the presheaf $x \mapsto H^p_{fppf}(x, \mathcal{F})$, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images}. To prove (1) it suffices to show that $H^p_{fppf}(x, \mathcal{F}) = 0$ for $p > 0$ whenever $x$ lies over an affine scheme $U$. By Lemma \ref{lemma-cohomology-restriction} we have $H^p_{fppf}(x, \mathcal{F}) = H^p((\Sch/U)_{fppf}, x^{-1}\mathcal{F})$. Combining Descent, Lemma \ref{descent-lemma-quasi-coherent-and-flat-base-change} with Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero} we see that these cohomology groups are zero. \medskip\noindent We have seen above that $\epsilon_*\mathcal{F}$ and $\mathcal{F}$ are the sheaves on $\mathcal{X}_\etale$ and $\mathcal{X}_{fppf}$ corresponding to the same presheaf on $\mathcal{X}$ (and this is true more generally for any sheaf in the fppf topology on $\mathcal{X}$). We often abusively identify $\mathcal{F}$ and $\epsilon_*\mathcal{F}$ and this is the sense in which parts (2) and (3) of the lemma should be understood. Thus part (2) follows formally from (1) and the Leray spectral sequence, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-apply-Leray}. \medskip\noindent Finally we prove (3). The sheaf $R^if_*\mathcal{F}$ (resp.\ $Rf_{\etale, *}\mathcal{F}$) is the sheaf associated to the presheaf $$ y \longmapsto H^i_\tau\Big((\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}, \ \text{pr}^{-1}\mathcal{F}\Big) $$ where $\tau$ is $fppf$ (resp.\ $\etale$), see Lemma \ref{lemma-pushforward-restriction}. Note that $\text{pr}^{-1}\mathcal{F}$ satisfies properties (a) and (b) also (by Lemmas \ref{lemma-pullback-lqc} and \ref{lemma-comparison}), hence these two presheaves are equal by (2). This immediately implies (3).
This proof is a repeat of the proof in the case of schemes, see Properties, Proposition \ref{properties-proposition-coherator} and the case of algebraic spaces, see Properties of Spaces, Proposition \ref{spaces-properties-proposition-coherator}. We advise the reader to read either of those proofs first. \medskip\noindent Part (1) means $\QCoh(\mathcal{O}_\mathcal{X})$ (a) has all colimits, (b) filtered colimits are exact, and (c) has a generator, see Injectives, Section \ref{injectives-section-grothendieck-conditions}. By Lemma \ref{lemma-quasi-coherent-algebraic-stack} colimits in $\QCoh(\mathcal{O}_X)$ exist and agree with colimits in $\textit{Mod}(\mathcal{O}_X)$. By Modules on Sites, Lemma \ref{sites-modules-lemma-limits-colimits} filtered colimits are exact. Hence (a) and (b) hold. \medskip\noindent Choose a presentation $\mathcal{X} = [U/R]$ so that $(U, R, s, t, c)$ is a smooth groupoid in algebraic spaces and in particular $s$ and $t$ are flat morphisms of algebraic spaces. By Lemma \ref{lemma-quasi-coherent-algebraic-stack} above we have $\QCoh(\mathcal{O}_\mathcal{X}) = \QCoh(U, R, s, t, c)$. By Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-set-generators} there exists a set $T$ and a family $(\mathcal{F}_t)_{t \in T}$ of quasi-coherent sheaves on $\mathcal{X}$ such that every quasi-coherent sheaf on $\mathcal{X}$ is the directed colimit of its subsheaves which are isomorphic to one of the $\mathcal{F}_t$. Thus $\bigoplus_t \mathcal{F}_t$ is a generator of $\QCoh(\mathcal{O}_X)$ and we conclude that (c) holds. The assertions on limits and injectives hold in any Grothendieck abelian category, see Injectives, Theorem \ref{injectives-theorem-injective-embedding-grothendieck} and Lemma \ref{injectives-lemma-grothendieck-products}. \medskip\noindent Proof of (2). To construct $Q$ we use the following general procedure. Given an object $\mathcal{F}$ of $\textit{Mod}(\mathcal{O}_\mathcal{X})$ we consider the functor $$ \QCoh(\mathcal{O}_\mathcal{X})^{opp} \longrightarrow \textit{Sets}, \quad \mathcal{G} \longmapsto \Hom_\mathcal{X}(\mathcal{G}, \mathcal{F}) $$ This functor transforms colimits into limits, hence is representable, see Injectives, Lemma \ref{injectives-lemma-grothendieck-brown}. Thus there exists a quasi-coherent sheaf $Q(\mathcal{F})$ and a functorial isomorphism $\Hom_\mathcal{X}(\mathcal{G}, \mathcal{F}) = \Hom_\mathcal{X}(\mathcal{G}, Q(\mathcal{F}))$ for $\mathcal{G}$ in $\QCoh(\mathcal{O}_\mathcal{X})$. By the Yoneda lemma (Categories, Lemma \ref{categories-lemma-yoneda}) the construction $\mathcal{F} \leadsto Q(\mathcal{F})$ is functorial in $\mathcal{F}$. By construction $Q$ is a right adjoint to the inclusion functor. The fact that $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism when $\mathcal{F}$ is quasi-coherent is a formal consequence of the fact that the inclusion functor $\QCoh(\mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{O}_\mathcal{X})$ is fully faithful.
By Lemma \ref{lemma-check-exactness-covering} it suffices to check exactness after pulling back to $\mathcal{U}$. By Lemma \ref{lemma-base-change-cech-complex} the pullback of the extended relative {\v C}ech complex is isomorphic to the extend relative {\v C}ech complex for the morphism $\mathcal{U} \times_\mathcal{X} \mathcal{U} \to \mathcal{U}$ and an abelian sheaf on $\mathcal{U}_\tau$. Since there is a section $\Delta_{\mathcal{U}/\mathcal{X}} : \mathcal{U} \to \mathcal{U} \times_\mathcal{X} \mathcal{U}$ exactness follows from Lemma \ref{lemma-homotopy-sheafified}.
To see this we will check the hypotheses (1) -- (4) of Lemma \ref{lemma-cech-to-cohomology}. The $1$-morphism $f$ is faithful by Algebraic Stacks, Lemma \ref{algebraic-lemma-characterize-representable-by-algebraic-spaces}. This proves (4). Hypothesis (3) follows from the fact that $\mathcal{U}$ is an algebraic stack, see Lemma \ref{lemma-fibre-products}. To see (2) apply Lemma \ref{lemma-surjective-flat-locally-finite-presentation}. Condition (1) is satisfied by fiat.