hcju/stacksqa huggingface dataset:
This lemma is a trivial consequence of Lemmas \ref{lemma-sum-divisors-associated-Weil} and \ref{lemma-commutativity-effective-Cartier-proper-intersection} in case the sums are finite, e.g., if $X$ is quasi-compact. Hence we suggest the reader skip the proof. \medskip\noindent Here is the proof in the general case. Let $i_j : D_j \to X$ be the closed immersions Let $p : \coprod D_j \to X$ denote coproduct of the morphisms $i_j$. Let $\{Z_a\}_{a \in A}$ be the collection of irreducible components of $\bigcup D_j$. For each $j$ we write $$ [D_j]_{n - 1} = \sum d_{j, a}[Z_a]. $$ By Lemma \ref{lemma-sum-divisors-associated-Weil} we have $$ [D]_{n - 1} = \sum n_j d_{j, a} [Z_a], \quad [D']_{n - 1} = \sum m_j d_{j, a} [Z_a]. $$ By Lemma \ref{lemma-improved-additivity} we have $$ D \cdot [D']_{n - 1} = p_*\left(\sum n_j i_j^*[D']_{n - 1} \right), \quad D' \cdot [D]_{n - 1} = p_*\left(\sum m_{j'} i_{j'}^*[D]_{n - 1} \right). $$ As in the definition of the Gysin homomorphisms (see Chow Homology, Definition \ref{chow-definition-gysin-homomorphism}) we choose cycles $\beta_{a, j}$ on $D_j \cap Z_a$ representing $i_j^*[Z_a]$. (Note that in fact $\beta_{a, j} = [D_j \cap Z_a]_{n - 2}$ if $Z_a$ is not contained in $D_j$, i.e., there is no choice in that case.) Now since $p$ is a closed immersion when restricted to each of the $D_j$ we can (and we will) view $\beta_{a, j}$ as a cycle on $X$. Plugging in the formulas for $[D]_{n - 1}$ and $[D']_{n - 1}$ obtained above we see that $$ D \cdot [D']_{n - 1} = \sum\nolimits_{j, j', a} n_j m_{j'} d_{j', a} \beta_{a, j}, \quad D' \cdot [D]_{n - 1} = \sum\nolimits_{j, j', a} m_{j'} n_j d_{j, a} \beta_{a, j'}. $$ Moreover, with the same conventions we also have $$ D_j \cdot [D_{j'}]_{n - 1} = \sum d_{j', a} \beta_{a, j}. $$ In these terms Lemma \ref{lemma-commutativity-effective-Cartier-proper-intersection} (see also its proof) says that for $j \not = j'$ the cycles $\sum d_{j', a} \beta_{a, j}$ and $\sum d_{j, a} \beta_{a, j'}$ are equal as cycles! Hence we see that \begin{eqnarray*} D \cdot [D']_{n - 1} & = & \sum\nolimits_{j, j', a} n_j m_{j'} d_{j', a} \beta_{a, j} \\ & = & \sum\nolimits_{j \not = j'} n_j m_{j'} \left(\sum\nolimits_a d_{j', a} \beta_{a, j}\right) + \sum\nolimits_{j, a} n_j m_j d_{j, a} \beta_{a, j} \\ & = & \sum\nolimits_{j \not = j'} n_j m_{j'} \left(\sum\nolimits_a d_{j, a} \beta_{a, j'}\right) + \sum\nolimits_{j, a} n_j m_j d_{j, a} \beta_{a, j} \\ & = & \sum\nolimits_{j, j', a} m_{j'} n_j d_{j, a} \beta_{a, j'} \\ & = & D' \cdot [D]_{n - 1} \end{eqnarray*} and we win.
[First proof of Lemma \ref{lemma-commutativity-effective-Cartier}] First, let us prove this in case $X$ is quasi-compact. In this case, apply Lemma \ref{lemma-blowing-up-intersections} to $X$ and the two element set $\{D, D'\}$ of effective Cartier divisors. Thus we get a proper morphism $b : X' \to X$, a finite collection of effective Cartier divisors $D'_j \subset X'$ intersecting pairwise in codimension $\geq 2$, with $b^{-1}(D) = \sum n_j D'_j$, and $b^{-1}(D') = \sum m_j D'_j$. Note that $b_*[b^{-1}(D)]_{n - 1} = [D]_{n - 1}$ in $Z_{n - 1}(X)$ and similarly for $D'$, see Lemma \ref{lemma-push-pull-effective-Cartier}. Hence, by Chow Homology, Lemma \ref{chow-lemma-pushforward-cap-c1} we have $$ D \cdot [D']_{n - 1} = b_*\left(b^{-1}(D) \cdot [b^{-1}(D')]_{n - 1}\right) $$ in $\CH_{n - 2}(X)$ and similarly for the other term. Hence the lemma follows from the equality $b^{-1}(D) \cdot [b^{-1}(D')]_{n - 1} = b^{-1}(D') \cdot [b^{-1}(D)]_{n - 1}$ in $\CH_{n - 2}(X')$ of Lemma \ref{lemma-commutativity-effective-Cartier-proper-intersection-infinite}. \medskip\noindent Note that in the proof above, each referenced lemma works also in the general case (when $X$ is not assumed quasi-compact). The only minor change in the general case is that the morphism $b : U' \to U$ we get from applying Lemma \ref{lemma-blowing-up-intersections} has as its target an open $U \subset X$ whose complement has codimension $\geq 3$. Hence by Chow Homology, Lemma \ref{chow-lemma-restrict-to-open} we see that $\CH_{n - 2}(U) = \CH_{n - 2}(X)$ and after replacing $X$ by $U$ the rest of the proof goes through unchanged.
Let $t : C_i \to X$ be the closed immersion. Since $t$ is the inclusion of an effective Cartier divisor we conclude from Duality for Schemes, Lemmas \ref{duality-lemma-twisted-inverse-image-closed} and \ref{duality-lemma-sheaf-with-exact-support-effective-Cartier} that we have $t^!(\mathcal{L}) = \mathcal{L}(C_i)|_{C_i}$ for every invertible $\mathcal{O}_X$-module $\mathcal{L}$. Consider the commutative diagram $$ \xymatrix{ C_i \ar[r]_t \ar[d]_g & X \ar[d]^f \\ \Spec(k) \ar[r]^s & \Spec(R) } $$ Observe that $C_i$ is a Gorenstein curve (Semistable Reduction, Lemma \ref{models-lemma-gorenstein}) with invertible dualizing module $\omega_{C_i}$ characterized by the property $\omega_{C_i}[0] = g^!\mathcal{O}_{\Spec(k)}$. See Algebraic Curves, Lemma \ref{curves-lemma-duality-dim-1}, its proof, and Algebraic Curves, Lemmas \ref{curves-lemma-duality-dim-1-CM} and \ref{curves-lemma-rr}. On the other hand, $s^!(R[1]) = k$ and hence $$ \omega_{C_i}[0] = g^! s^!(R[1]) = t^!f^!(R[1]) = t^!\omega_X $$ Combining the above we obtain the statement of the lemma.
Let $\mathcal{H}$ be a presheaf on $\mathcal{Z}$. Then $(g \circ f)^p\mathcal{H} = f^p (g^p\mathcal{H})$ is given by the equalities $$ (g \circ f)^p\mathcal{H}(x) = \mathcal{H}((g \circ f)(x)) = \mathcal{H}(g(f(x))) = f^p (g^p\mathcal{H})(x). $$ We omit the verification that this is compatible with restriction maps. \medskip\noindent Next, we define the transformation ${}_p(g \circ f) \to {}_pg \circ {}_pf$. Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$. If $z$ is an object of $\mathcal{Z}$ then we get a category $\mathcal{J}$ of quadruples $(x, f(x) \to y, y, g(y) \to z)$ and a category $\mathcal{I}$ of pairs $(x, g(f(x)) \to z)$. There is a canonical functor $\mathcal{J} \to \mathcal{I}$ sending the object $(x, \alpha : f(x) \to y, y, \beta : g(y) \to z)$ to $(x, \beta \circ f(\alpha) : g(f(x)) \to z)$. This gives the arrow in \begin{align*} ({}_p(g \circ f)\mathcal{F})(z) & = \lim_{g(f(x)) \to z} \mathcal{F}(x) \\ & = \lim_\mathcal{I} \mathcal{F} \\ & \to \lim_\mathcal{J} \mathcal{F} \\ & = \lim_{g(y) \to z} \Big(\lim_{f(x) \to y} \mathcal{F}(x)\Big) \\ & = ({}_pg \circ {}_pf\mathcal{F})(x) \end{align*} by Categories, Lemma \ref{categories-lemma-functorial-limit}. We omit the verification that this is compatible with restriction maps. An alternative to this direct construction is to define ${}_p(g \circ f) \cong {}_pg \circ {}_pf$ as the unique map compatible with the adjointness properties. This also has the advantage that one does not need to prove the compatibility. \medskip\noindent Compatibility with adjointness of $(f^p, {}_pf)$, $(g^p, {}_pg)$, and $((g \circ f)^p, {}_p(g \circ f))$ means that given presheaves $\mathcal{H}$ and $\mathcal{F}$ as above we have a commutative diagram $$ \xymatrix{ \Mor_{\textit{PSh}(\mathcal{X})}(f^pg^p\mathcal{H}, \mathcal{F}) \ar@{=}[r] \ar@{=}[d] & \Mor_{\textit{PSh}(\mathcal{Y})}(g^p\mathcal{H}, {}_pf\mathcal{F}) \ar@{=}[r] & \Mor_{\textit{PSh}(\mathcal{Y})}(\mathcal{H}, {}_pg{}_pf\mathcal{F}) \\ \Mor_{\textit{PSh}(\mathcal{X})}((g \circ f)^p\mathcal{G}, \mathcal{F}) \ar@{=}[rr] & & \Mor_{\textit{PSh}(\mathcal{Y})}(\mathcal{G}, {}_p(g \circ f)\mathcal{F}) \ar[u] } $$ Proof omitted.
Let $\mathcal{G}$ be a presheaf on $\mathcal{Y}$. Then $t^p : g^p\mathcal{G} \to f^p\mathcal{G}$ is given by the family of maps $$ g^p\mathcal{G}(x) = \mathcal{G}(g(x)) \xrightarrow{\mathcal{G}(t_x)} \mathcal{G}(f(x)) = f^p\mathcal{G}(x) $$ parametrized by $x \in \Ob(\mathcal{X})$. This makes sense as $t_x : f(x) \to g(x)$ and $\mathcal{G}$ is a contravariant functor. We omit the verification that this is compatible with restriction mappings. \medskip\noindent To define the transformation ${}_pt$ for $y \in \Ob(\mathcal{Y})$ define ${}_y^f\mathcal{I}$, resp.\ ${}_y^g\mathcal{I}$ to be the category of pairs $(x, \psi : f(x) \to y)$, resp.\ $(x, \psi : g(x) \to y)$, see Sites, Section \ref{sites-section-more-functoriality-PSh}. Note that $t$ defines a functor ${}_yt : {}_y^g\mathcal{I} \to {}_y^f\mathcal{I}$ given by the rule $$ (x, g(x) \to y) \longmapsto (x, f(x) \xrightarrow{t_x} g(x) \to y). $$ Note that for $\mathcal{F}$ a presheaf on $\mathcal{X}$ the composition of ${}_yt$ with $\mathcal{F} : {}_y^f\mathcal{I}^{opp} \to \textit{Sets}$, $(x, f(x) \to y) \mapsto \mathcal{F}(x)$ is equal to $\mathcal{F} : {}_y^g\mathcal{I}^{opp} \to \textit{Sets}$. Hence by Categories, Lemma \ref{categories-lemma-functorial-limit} we get for every $y \in \Ob(\mathcal{Y})$ a canonical map $$ ({}_pf\mathcal{F})(y) = \lim_{{}_y^f\mathcal{I}} \mathcal{F} \longrightarrow \lim_{{}_y^g\mathcal{I}} \mathcal{F} = ({}_pg\mathcal{F})(y) $$ We omit the verification that this is compatible with restriction mappings. An alternative to this direct construction is to define ${}_pt$ as the unique map compatible with the adjointness properties of the pairs $(f^p, {}_pf)$ and $(g^p, {}_pg)$ (see below). This also has the advantage that one does not need to prove the compatibility. \medskip\noindent Compatibility with adjointness of $(f^p, {}_pf)$ and $(g^p, {}_pg)$ means that given presheaves $\mathcal{G}$ and $\mathcal{F}$ as above we have a commutative diagram $$ \xymatrix{ \Mor_{\textit{PSh}(\mathcal{X})}(f^p\mathcal{G}, \mathcal{F}) \ar@{=}[r] \ar[d]_{- \circ t^p} & \Mor_{\textit{PSh}(\mathcal{Y})}(\mathcal{G}, {}_pf\mathcal{F}) \ar[d]^{{}_pt \circ -} \\ \Mor_{\textit{PSh}(\mathcal{X})}(g^p\mathcal{G}, \mathcal{F}) \ar@{=}[r] & \Mor_{\textit{PSh}(\mathcal{Y})}(\mathcal{G}, {}_pg\mathcal{F}) } $$ Proof omitted. Hint: Work through the proof of Sites, Lemma \ref{sites-lemma-adjoints-pu} and observe the compatibility from the explicit description of the horizontal and vertical maps in the diagram. \medskip\noindent We omit the verification that this is compatible with vertical and horizontal compositions. Hint: The proof of this for $t^p$ is straightforward and one can conclude that this holds for the ${}_pt$ maps using compatibility with adjointness.
Denote $p : \mathcal{X} \to (\Sch/S)_{fppf}$ and $q : \mathcal{Y} \to (\Sch/S)_{fppf}$ the structural functors. Then $q = p \circ f$, hence $q^{-1} = f^{-1} \circ p^{-1}$ by Lemma \ref{lemma-1-morphisms-presheaves}. The result follows.
By Sites, Lemma \ref{sites-lemma-enough-points-local} we have to show that there exists a family of objects $x$ of $\mathcal{X}$ such that $\mathcal{X}_\tau/x$ has enough points and such that the sheaves $h_x^\#$ cover the final object of the category of sheaves. By Lemma \ref{lemma-localizing} and \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-points-fppf} we see that $\mathcal{X}_\tau/x$ has enough points for every object $x$ and we win.
Choose an equivalence $j : \mathcal{S}_F \to \mathcal{X}$, see Algebraic Stacks, Sections \ref{algebraic-section-split} and \ref{algebraic-section-representable-by-algebraic-spaces}. An object of $F_\etale$ is a scheme $U$ together with an \'etale morphism $\varphi : U \to F$. Then $\varphi$ is an object of $\mathcal{S}_F$ over $U$. Hence $j(\varphi)$ is an object of $\mathcal{X}$ over $U$. In this way $j$ induces a functor $u : F_\etale \to \mathcal{X}$. It is clear that $u$ is continuous and cocontinuous for the \'etale topology on $\mathcal{X}$. Since $j$ is an equivalence, the functor $u$ is fully faithful. Also, fibre products and equalizers exist in $F_\etale$ and $u$ commutes with them because these are computed on the level of underlying schemes in $F_\etale$. Thus Sites, Lemmas \ref{sites-lemma-when-shriek}, \ref{sites-lemma-preserve-equalizers}, and \ref{sites-lemma-back-and-forth} apply. In particular $u$ defines a morphism of topoi $i_F : \Sh(F_\etale) \to \Sh(\mathcal{X}_\etale)$ and there exists a left adjoint $i_{F, !}$ of $i_F^{-1}$ which commutes with fibre products and equalizers. \medskip\noindent We claim that $i_{F, !}$ is exact. If this is true, then we can define $\pi_F$ by the rules $\pi_F^{-1} = i_{F, !}$ and $\pi_{F, *} = i_F^{-1}$ and everything is clear. To prove the claim, note that we already know that $i_{F, !}$ is right exact and preserves fibre products. Hence it suffices to show that $i_{F, !}* = *$ where $*$ indicates the final object in the category of sheaves of sets. Let $U$ be a scheme and let $\varphi : U \to F$ be surjective and \'etale. Set $R = U \times_F U$. Then $$ \xymatrix{ h_R \ar@<1ex>[r] \ar@<-1ex>[r] & h_U \ar[r] & {*} } $$ is a coequalizer diagram in $\Sh(F_\etale)$. Using the right exactness of $i_{F, !}$, using $i_{F, !} = (u_p\ )^\#$, and using Sites, Lemma \ref{sites-lemma-pullback-representable-presheaf} we see that $$ \xymatrix{ h_{u(R)} \ar@<1ex>[r] \ar@<-1ex>[r] & h_{u(U)} \ar[r] & i_{F, !}{*} } $$ is a coequalizer diagram in $\Sh(F_\etale)$. Using that $j$ is an equivalence and that $F = U/R$ it follows that the coequalizer in $\Sh(\mathcal{X}_\etale)$ of the two maps $h_{u(R)} \to h_{u(U)}$ is $*$. We omit the proof that these morphisms are compatible with structure sheaves.
By Lemma \ref{lemma-pullback-quasi-coherent} the condition is necessary. Conversely, since $x^*\mathcal{F}$ is just the restriction to $\mathcal{X}_{fppf}/x$ we see that it is sufficient directly from the definition of a quasi-coherent sheaf (and the fact that the notion of being quasi-coherent is an intrinsic property of sheaves of modules, see Modules on Sites, Section \ref{sites-modules-section-intrinsic}).
Assume either (1), (2), or (3) holds. Let $x$ be an object of $\mathcal{X}$ lying over the scheme $U$. Recall that $x^*\mathcal{F} = x^{-1}\mathcal{F}$ is just the restriction to $\mathcal{X}/x = (\Sch/U)_\tau$ where $\tau = fppf$, $\tau = \etale$, or $\tau = Zar$, see Section \ref{section-restriction}. By the definition of quasi-coherent modules on a ringed site this restriction is quasi-coherent provided $\mathcal{F}$ is. By Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent} we see that $x^*\mathcal{F}$ is the sheaf associated to a quasi-coherent $\mathcal{O}_U$-module and is therefore a quasi-coherent module in the fppf, \'etale, and Zariski topology; here we also use Descent, Lemma \ref{descent-lemma-sheaf-condition-holds} and Definition \ref{descent-definition-structure-sheaf}. Since this holds for every object $x$ of $\mathcal{X}$, we see that $\mathcal{F}$ is a sheaf in any of the three topologies. Moreover, we find that $\mathcal{F}$ is quasi-coherent in any of the three topologies directly from the definition of being quasi-coherent and the fact that $x$ is an arbitrary object of $\mathcal{X}$.
Assume $\mathcal{F}$ is quasi-coherent. Then $\mathcal{F}$ is a sheaf for the fppf topology, hence a sheaf for the \'etale topology. Moreover, any pullback of $\mathcal{F}$ to a ringed topos is quasi-coherent, hence the restrictions $x^*\mathcal{F}|_{U_\etale}$ are quasi-coherent. This proves $\mathcal{F}$ is locally quasi-coherent. Let $y$ be an object of $\mathcal{X}$ with $V = p(y)$. We have seen that $\mathcal{X}/y = (\Sch/V)_{fppf}$. By Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent} it follows that $y^*\mathcal{F}$ is the quasi-coherent module associated to a (usual) quasi-coherent module $\mathcal{F}_V$ on the scheme $V$. Hence certainly the comparison maps (\ref{equation-comparison-modules}) are isomorphisms. \medskip\noindent Conversely, suppose that $\mathcal{F}$ satisfies (1) and (2). Let $y$ be an object of $\mathcal{X}$ with $V = p(y)$. Denote $\mathcal{F}_V$ the quasi-coherent module on the scheme $V$ corresponding to the restriction $y^*\mathcal{F}|_{V_\etale}$ which is quasi-coherent by assumption (1), see Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent}. Condition (2) now signifies that the restrictions $x^*\mathcal{F}|_{U_\etale}$ for $x$ over $y$ are each isomorphic to the (\'etale sheaf associated to the) pullback of $\mathcal{F}_V$ via the corresponding morphism of schemes $U \to V$. Hence $y^*\mathcal{F}$ is the sheaf on $(\Sch/V)_{fppf}$ associated to $\mathcal{F}_V$. Hence it is quasi-coherent (by Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent} again) and we see that $\mathcal{F}$ is quasi-coherent on $\mathcal{X}$ by Lemma \ref{lemma-characterize-quasi-coherent}.
Each of these statements follows from the corresponding statement of Descent, Lemma \ref{descent-lemma-equivalence-quasi-coherent-limits}. For example, suppose that $\mathcal{I} \to \textit{LQCoh}(\mathcal{O}_\mathcal{X})$, $i \mapsto \mathcal{F}_i$ is a diagram. Consider the object $\mathcal{F} = \colim_i \mathcal{F}_i$ of $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$. For any object $x$ of $\mathcal{X}$ with $U = p(x)$ the pullback functor $x^*$ commutes with all colimits as it is a left adjoint. Hence $x^*\mathcal{F} = \colim_i x^*\mathcal{F}_i$. Similarly we have $x^*\mathcal{F}|_{U_\etale} = \colim_i x^*\mathcal{F}_i|_{U_\etale}$. Now by assumption each $x^*\mathcal{F}_i|_{U_\etale}$ is quasi-coherent, hence the colimit is quasi-coherent by the aforementioned Descent, Lemma \ref{descent-lemma-equivalence-quasi-coherent-limits}. This proves (1). \medskip\noindent It follows from (1) that cokernels exist in $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ and agree with the cokernels computed in $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of $\textit{LQCoh}(\mathcal{O}_\mathcal{X})$ and let $\mathcal{K} = \Ker(\varphi)$ computed in $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$. If we can show that $\mathcal{K}$ is a locally quasi-coherent module, then the proof of (2) is complete. To see this, note that kernels are computed in the category of presheaves (no sheafification necessary). Hence $\mathcal{K}|_{U_\etale}$ is the kernel of the map $\mathcal{F}|_{U_\etale} \to \mathcal{G}|_{U_\etale}$, i.e., is the kernel of a map of quasi-coherent sheaves on $U_\etale$ whence quasi-coherent by Descent, Lemma \ref{descent-lemma-equivalence-quasi-coherent-limits}. This proves (2). \medskip\noindent Parts (3), (4), and (5) follow in exactly the same way. Details omitted.
Let $\mathcal{I} \to \QCoh(\mathcal{O}_\mathcal{X})$, $i \mapsto \mathcal{F}_i$ be a diagram. Viewing $\mathcal{F}_i$ as quasi-coherent modules in the Zariski topology (Lemma \ref{lemma-characterize-quasi-coherent-bis}), we may consider the object $\mathcal{F} = \colim_i \mathcal{F}_i$ of $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal{X})$. For any object $x$ of $\mathcal{X}$ with $U = p(x)$ the restriction functor $x^*$ (Section \ref{section-restriction}) commutes with all colimits as it is a left adjoint. Hence $x^*\mathcal{F} = \colim_i x^*\mathcal{F}_i$ in $\textit{Mod}((\Sch/U)_{Zar}, \mathcal{O})$. Observe that $x_i^*\mathcal{F}_i$ is a quasi-coherent object (because restrictions of quasi-coherent modules are quasi-coherent). Thus by the equivalence in Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent} and by the compatibility with colimits in Descent, Lemma \ref{descent-lemma-equivalence-quasi-coherent-limits} we conclude that $x^*\mathcal{F}$ is quasi-coherent. Thus $\mathcal{F}$ is quasi-coherent, see Lemma \ref{lemma-characterize-quasi-coherent-bis}. Thus we see that $\QCoh(\mathcal{O}_\mathcal{X})$ has colimits and they agree with colimits in the category $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal{X})$. Since the other categories listed are full subcategories of $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal{X})$ we conclude part (1) holds. \medskip\noindent Parts (2) and (3) are proved in the same way. Details omitted.
We may assume $\mathcal{Y}$ is the stackification of $\mathcal{X}$. We claim that $f : \mathcal{X} \to \mathcal{Y}$ is a special cocontinuous functor, see Sites, Definition \ref{sites-definition-special-cocontinuous-functor} which will prove the lemma. By Stacks, Lemma \ref{stacks-lemma-topology-inherited-functorial} the functor $f$ is continuous and cocontinuous. By Stacks, Lemma \ref{stacks-lemma-stackify} we see that conditions (3), (4), and (5) of Sites, Lemma \ref{sites-lemma-equivalence} hold.
We may assume $\mathcal{Y}$ is the stackification of $\mathcal{X}$. The first assertion is clear from Lemma \ref{lemma-stackification} and $\mathcal{O}_\mathcal{X} = f^{-1}\mathcal{O}_\mathcal{Y}$. Pullback of quasi-coherent sheaves are quasi-coherent, see Lemma \ref{lemma-pullback-quasi-coherent}. Hence it suffices to show that if $f^*\mathcal{G}$ is quasi-coherent, then $\mathcal{G}$ is. To see this, let $y$ be an object of $\mathcal{Y}$. Translating the condition that $\mathcal{Y}$ is the stackification of $\mathcal{X}$ we see there exists an fppf covering $\{y_i \to y\}$ in $\mathcal{Y}$ such that $y_i \cong f(x_i)$ for some $x_i$ object of $\mathcal{X}$. Say $x_i$ and $y_i$ lie over the scheme $U_i$. Then $f^*\mathcal{G}$ being quasi-coherent, means that $x_i^*f^*\mathcal{G}$ is quasi-coherent. Since $x_i^*f^*\mathcal{G}$ is isomorphic to $y_i^*\mathcal{G}$ (as sheaves on $(\Sch/U_i)_{fppf}$ we see that $y_i^*\mathcal{G}$ is quasi-coherent. It follows from Modules on Sites, Lemma \ref{sites-modules-lemma-local-final-object} that the restriction of $\mathcal{G}$ to $\mathcal{Y}/y$ is quasi-coherent. Hence $\mathcal{G}$ is quasi-coherent by Lemma \ref{lemma-characterize-quasi-coherent}.
According to Lemma \ref{lemma-stackification-quasi-coherent} the stackification map $[U/_{\!p}R] \to [U/R]$ (see Groupoids in Spaces, Definition \ref{spaces-groupoids-definition-quotient-stack}) induces an equivalence of categories of quasi-coherent sheaves and of fppf $\mathcal{O}$-modules. Thus it suffices to prove the lemma with $\mathcal{X} = [U/_{\!p}R]$. By Proposition \ref{proposition-quasi-coherent} and its proof there exists a quasi-coherent module $(\mathcal{G}, \alpha)$ on $(U, R, s, t, c)$ such that $\mathcal{F}$ is given by the rule $\mathcal{F}(T, u) = \Gamma(T, u^*\mathcal{G})$. In particular $x^*\mathcal{F}|_{U_\etale} = \mathcal{G}$ and it is clear that the map of the statement of the lemma is injective. Moreover, given a map $\varphi : \mathcal{G} \to x^*\mathcal{H}|_{U_\etale}$ and given any object $y = (T, u)$ of $[U/_{\!p}R]$ we can consider the map $$ \mathcal{F}(y) = \Gamma(T, u^*\mathcal{G}) \xrightarrow{u_{small}^*\varphi} \Gamma(T, u_{small}^*x^*\mathcal{H}|_{U_\etale}) \rightarrow \Gamma(T, y^*\mathcal{H}|_{T_\etale}) = \mathcal{H}(y) $$ where the second arrow is the comparison map (\ref{equation-comparison-modules}) for the sheaf $\mathcal{H}$. This assignment is compatible with the restriction mappings of the sheaves $\mathcal{F}$ and $\mathcal{G}$ for morphisms of $[U/_{\!p}R]$ if the cocycle condition of the lemma is satisfied. Proof omitted. Hint: the restriction maps of $\mathcal{F}$ are made explicit in terms of $(\mathcal{G}, \alpha)$ in the proof of Proposition \ref{proposition-quasi-coherent}.
The first statement follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-of-open} and the equivalence of  Lemma \ref{lemma-localizing-structure-sheaf}. The second statement follows from the first combined with \'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-cohomology-big-small}.