hcju/stacksqa huggingface dataset:
By Proposition \ref{proposition-trace-well-defined}, we may assume we have a bounded complex $P^\bullet$ of filtered finite projectives of $\text{Fil}^f(\text{Mod}_\Lambda)$ and a map $f^\bullet : P^\bullet\to P^\bullet$ in $\text{Comp}(\text{Fil}^f(\text{Mod}_\Lambda))$. So the lemma follows from the following result, which proof is left to the reader.
[Sketch of proof] Observe first that the assumption makes sense because $H^i(C, \underline{\mathbf{Z}/\ell^n \mathbf{Z}})$ is a free $\mathbf{Z}/\ell^n \mathbf{Z}$-module for all $i$. The trace of $\varphi^*$ on the 0th degree cohomology is 1. The choice of a primitive $\ell^n$th root of unity in $k$ gives an isomorphism $$ H^i(C, \underline{\mathbf{Z}/\ell^n \mathbf{Z}}) \cong H^i(C, \mu_{\ell^n}) $$ compatibly with the action of the geometric Frobenius. On the other hand, $H^1(C, \mu_{\ell^n}) = J[\ell^n]$. Therefore, \begin{eqnarray*} \text{Tr}(\varphi^* |_{H^1 (C, \underline{\mathbf{Z}/\ell^n \mathbf{Z}})})) & = & \text{Tr}_J (\varphi^*) \mod \ell^n \\ & = & \text{Tr}_{\mathbf{Z}/\ell^n \mathbf{Z}} (\varphi^* : J[\ell^n] \to J[\ell^n]). \end{eqnarray*} Moreover, $H^2(C, \mu_{\ell^n}) = \Pic(C)/\ell^n\Pic(C) \cong \mathbf{Z}/\ell^n \mathbf{Z}$ where $\varphi^*$ is multiplication by $\deg \varphi$. Hence $$ \text{Tr} (\varphi^*|_{H^2 (C, \underline{\mathbf{Z}/\ell^n \mathbf{Z}})}) = \deg \varphi. $$ Thus we have $$ \sum_{i = 0}^2 (-1)^i \text{Tr}(\varphi^* |_{H^i (C, \underline{\mathbf{Z}/\ell^n \mathbf{Z}})}) = 1 - \text{Tr}_J(\varphi^*) + \deg \varphi \mod \ell^n $$ and the corollary follows from Theorem \ref{theorem-weil-trace-formula}.
Let $\Phi = \left\{\varphi_n\right\}_{n\geq 1} : \left\{\mathcal{F}_n\right\} \to \left\{\mathcal{G}_n\right\}$ be a morphism of $\mathbf{Z}_\ell$-sheaves. Set $$ \Coker(\Phi) = \left\{ \Coker\left(\mathcal{F}_n \xrightarrow{\varphi_n} \mathcal{G}_n\right) \right\}_{n\geq 1} $$ and $\Ker(\Phi)$ is the result of Lemma \ref{lemma-eventually-constant} applied to the inverse system $$ \left\{ \bigcap_{m\geq n} \Im\left(\Ker(\varphi_m) \to \Ker(\varphi_n)\right) \right\}_{n \geq 1}. $$ That this defines an abelian category is left to the reader.
[Proof of Lemma \ref{lemma-identify-h2c}] Let $Y\to^{\varphi}X$ be the finite \'etale Galois covering corresponding to $\Ker(\rho) \subset \pi_1(X, \overline\eta)$. So $$ \text{Aut}(Y/X)=Ind(\rho) $$ is Galois group. Then $\varphi^*\mathcal{F}_\rho =\underline M_Y$ and $$ \varphi_*\varphi^*\mathcal{F}_\rho\to \mathcal{F}_\rho $$ which gives \begin{align*} & H_c^2(X_{\overline{k}}, \varphi_*\varphi^*\mathcal{F}_\rho) \to H_c^2(X_{\overline{k}}, \mathcal{F}_\rho)\\ & =H_c^2(Y_{\overline{k}}, \varphi^*\mathcal{F}_\rho)\\ & =H_c^2(Y_{\overline{k}}, \underline M) = \oplus_{\text{irred. comp. of } \atop Y_{\overline{k}}}M \end{align*} $$ \Im(\rho) \to H_c^2(Y_{\overline{k}}, \underline M) = \oplus_{\text{irred. comp. of } \atop Y_{\overline{k}}} M \to_{\Im(\rho) \text{equivalent}} H_c^2(X_{\overline{k}}, \mathcal{F}_{\rho}) \to^{\text{trivial } \Im(\rho) \atop \text{action}} $$ irreducible curve $C/\overline{k}$, $H_c^2(C, \underline M)=M$. \medskip\noindent Since $$ {\text{set of irreducible } \atop \text{components of }Y_k} = \frac{Im(\rho)}{Im(\rho|_{\pi_1(X_{\overline{k}}, \overline \eta)})} $$ We conclude that $H_c^2(X_{\overline{k}}, \mathcal{F}_\rho)$ is a quotient of $M_{\pi_1(X_{\overline{k}}, \overline \eta)}$. On the other hand, there is a surjection $$ \mathcal{F}_\rho\to \mathcal{F}'' = {\text{ sheaf on } X\text{ associated to } \atop (M)_{\pi_1(X_{\overline{k}}, \overline \eta)}\leftarrow\pi_1(X, \overline \eta)} $$ $$ H_c^2(X_{\overline{k}}, \mathcal{F}_\rho)\to M_{\pi_1(X_{\overline{k}}, \overline\eta)} $$ The twist in Galois action comes from the fact that $H_c^2(X_{\overline{k}}, \mu_n)=^{\text{can}} \mathbf{Z}/n\mathbf{Z}$.
To prove the switching lemma use Theorem \ref{theorem-drinfeld-make-f} to obtain $f\in C(\overline{\mathbf{Q}}_l)$ eigenform ass. to $\rho$. Next, use Proposition \ref{proposition-cusp-forms-finite} to see that we may choose $f\in C(E')$ with $E \subset E'$ finite. Next we may complete $E'$ to see that we get $f\in C(E'_{\lambda'})$ eigenform with $E'_{\lambda'}$ a finite extension of $E_{\lambda'}$. And finally we use Theorem \ref{theorem-drinfeld-make-rho} to obtain $\rho': \pi_1(X) \to SL_2(E_{\lambda'}')$ abs. irred. and continuous after perhaps enlarging $E'_{\lambda'}$ a bit again.
We will use Derived Categories, Lemma \ref{derived-lemma-morphisms-from-projective-complex} without further mention in this proof. Let $P^\bullet$ be a bounded complex of finite projective $\Lambda$-modules and let $\alpha : P^\bullet \to K$ be an isomorphism in $D(\Lambda)$. Then $\alpha^{-1}\circ f\circ \alpha$ corresponds to a morphism of complexes $f^\bullet : P^\bullet \to P^\bullet$ well defined up to homotopy. Set $$ \text{Tr}(f) = \sum_i (-1)^i \text{Tr}(f^i : P^i \to P^i) \in \Lambda^\natural. $$ Given $P^\bullet$ and $\alpha$, this is independent of the choice of $f^\bullet$. Namely, any other choice is of the form $\tilde{f}^\bullet = f^\bullet + dh +hd$ for some $h^i : P^i \to P^{i-1}(i\in \mathbf{Z})$. But \begin{eqnarray*} \text{Tr}(dh) & = & \sum_i (-1)^i \text{Tr}(P^i\xrightarrow{dh} P^i) \\ & = & \sum_i (-1)^i \text{Tr}(P^{i-1}\xrightarrow{hd} P^{i-1}) \\ & = & -\sum_i (-1)^{i-1}\text{Tr}(P^{i-1}\xrightarrow{hd} P^{i-1}) \\ & = & - \text{Tr}(hd) \end{eqnarray*} and so $\sum_i (-1)^i \text{Tr} ((dh+hd)|_{P^i})=0$. Furthermore, this is independent of the choice of $(P^\bullet , \alpha)$: suppose $(Q^\bullet, \beta)$ is another choice. The compositions $$ Q^\bullet \xrightarrow{\beta} K \xrightarrow{\alpha^{-1}} P^\bullet \quad\text{and}\quad P^\bullet \xrightarrow{\alpha} K \xrightarrow{\beta^{-1}} Q^\bullet $$ are representable by morphisms of complexes $\gamma_1^\bullet$ and $\gamma_2^\bullet$ respectively, such that $\gamma_1^\bullet \circ \gamma_2^\bullet$ is homotopic to the identity. Thus, the morphism of complexes $\gamma_2^\bullet\circ f^\bullet\circ \gamma_1^\bullet : Q^\bullet\to Q^\bullet$ represents the morphism $\beta^{-1}\circ f\circ\beta$ in $D(\Lambda)$. Now \begin{eqnarray*} \text{Tr}(\gamma_2^\bullet\circ f^\bullet\circ\gamma_1^\bullet|_{Q^\bullet}) & = & \text{Tr}(\gamma_1^\bullet \circ\gamma_2^\bullet \circ f^\bullet|_{P^\bullet})\\ & = & \text{Tr}(f^\bullet|_{P^\bullet}) \end{eqnarray*} by the fact that $\gamma_1^\bullet \circ \gamma_2^\bullet$ is homotopic to the identity and the independence of the choice of $f^\bullet$ we saw above.
[Sketch of proof.] The first step is to show: \begin{enumerate} \item[(1)] {\it The cohomology of $R\Gamma(X_{\bar k}, K)$ is bounded.} \end{enumerate} Consider the spectral sequence $$ H^i(X_{\bar k}, \underline H^j(K)) \Rightarrow H^{i+j} (R\Gamma(X_{\bar k}, K)). $$ Since $K$ is bounded and $\Lambda$ is finite, the sheaves $\underline H^j(K)$ are torsion. Moreover, $X_{\bar k}$ has finite cohomological dimension, so the left-hand side is nonzero for finitely many $i$ and $j$ only. Therefore, so is the right-hand side. \begin{enumerate} \item[(2)] {\it The cohomology groups $H^{i+j} (R\Gamma(X_{\bar k}, K))$ are finite.} \end{enumerate} Since the sheaves $\underline H^j(K)$ are constructible, the groups $H^i(X_{\bar k}, \underline H^j(K))$ are finite (\'Etale Cohomology, Section \ref{etale-cohomology-section-vanishing-torsion}) so it follows by the spectral sequence again. \begin{enumerate} \item[(3)] {\it $R\Gamma(X_{\bar k}, K)$ has finite $\text{Tor}$-dimension.} \end{enumerate} Let $N$ be a right $\Lambda$-module (in fact, since $\Lambda$ is finite, it suffices to assume that $N$ is finite). By the projection formula (change of module), $$ N \otimes^\mathbf{L}_\Lambda R \Gamma(X_{\bar k}, K) = R\Gamma(X_{\bar k}, N \otimes^\mathbf{L}_\Lambda K). $$ Therefore, $$ H^i (N \otimes^\mathbf{L}_\Lambda R\Gamma(X_{\bar k}, K)) = H^i(R\Gamma(X_{\bar k}, N \otimes_{\Lambda}^\mathbf{L} K)). $$ Now consider the spectral sequence $$ H^i (X_{\bar k}, \underline H^j (N \otimes_{\Lambda}^\mathbf{L} K)) \Rightarrow H^{i+j}(R\Gamma(X_{\bar k}, N \otimes_{\Lambda}^\mathbf{L} K)). $$ Since $K$ has finite $\text{Tor}$-dimension, $\underline H^j (N \otimes_{\Lambda}^\mathbf{L} K)$ vanishes universally for $j$ small enough, and the left-hand side vanishes whenever $i < 0$. Therefore $R\Gamma(X_{\bar k}, K)$ has finite $\text{Tor}$-dimension, as claimed. So it is a perfect complex by Lemma \ref{lemma-characterize-perfect}.
Assume that $\mathcal{F}$ is a sheaf. Then $\mathcal{F}(U)$ is the equalizer of the diagram above and by assumption we see $F(\mathcal{F}(U))$ is the equalizer of the corresponding diagram of sets. Hence $F(\mathcal{F})$ is a sheaf of sets. \medskip\noindent Assume that $F(\mathcal{F})$ is a sheaf. Let $E \in \Ob(\mathcal{C})$ be the equalizer of the two parallel arrows in Definition \ref{definition-sheaf-values-in-category}. We get a canonical morphism $\mathcal{F}(U) \to E$, simply because $\mathcal{F}$ is a presheaf. By assumption, the induced map $F(\mathcal{F}(U)) \to F(E)$ is an isomorphism, because $F(E)$ is the equalizer of the corresponding diagram of sets. Hence we see $\mathcal{F}(U) \to E$ is an isomorphism by condition (3) of the lemma.
Let $U$ be an open subset of $X$. Consider the diagram of (underlying) sets $$ \xymatrix{ \mathcal{F}(U) \ar[r] \ar[d] & \prod_{x \in U} \mathcal{F}_x \ar[d] \\ \mathcal{G}(U) \ar[r] & \prod_{x \in U} \mathcal{G}_x } $$ By assumption, and previous results, all but the left vertical arrow are morphisms of algebraic structures. In addition the bottom horizontal arrow is injective, see Lemma \ref{lemma-sheaf-subset-stalks}. Hence we conclude by Lemma \ref{lemma-image-contained-in}, see also Example \ref{example-application-lemma-image-contained-in}
Clearly, there is a commutative diagram $$ \xymatrix{ \mathcal{F} \ar[r] \ar[d] & \mathcal{F}^\# \ar[r] \ar[d] & \Pi(\mathcal{F}) \ar[d] \\ \mathcal{G} \ar[r] & \mathcal{G}^\# \ar[r] & \Pi(\mathcal{G}) \\ } $$ So it suffices to prove that $\mathcal{G} = \mathcal{G}^\#$. To see this it suffices to prove, for every point $x \in X$ the map $\mathcal{G}_x \to \mathcal{G}^\#_x$ is bijective, by Lemma \ref{lemma-points-exactness}. And this is Lemma \ref{lemma-stalk-sheafification} above.
It is clear that the diagram commutes. We have to show it is a fibre product diagram. The bottom horizontal arrow is injective since all the maps $\mathcal{F}_x \to \Pi(\mathcal{F})_x$ are injective (see beginning proof of Lemma \ref{lemma-stalk-sheafification}). A section $s \in \Pi(\mathcal{F})(U)$ is in $\mathcal{F}^\#$ if and only if $(*)$ holds. But $(*)$ says that around every point the section $s$ comes from a section of $\mathcal{F}$. By definition of the stalk functors, this is equivalent to saying that the value of $s$ in every stalk $\Pi(\mathcal{F})_x$ comes from an element of the stalk $\mathcal{F}_x$. Hence the lemma.
Recall the sheaf of sets $\Pi(\mathcal{F})$ defined in Section \ref{section-sheafification}. All the stalks $\mathcal{F}_x$ are abelian groups, see Lemma \ref{lemma-stalk-abelian-presheaf}. Hence $\Pi(\mathcal{F})$ is a sheaf of abelian groups by Example \ref{example-sheaf-product-pointwise-algebraic-structure}. Also, it is clear that the map $\mathcal{F} \to \Pi(\mathcal{F})$ is a morphism of abelian presheaves. If we show that condition $(*)$ of Section \ref{section-sheafification} defines a subgroup of $\Pi(\mathcal{F})(U)$ for all open subsets $U \subset X$, then $\mathcal{F}^\#$ canonically inherits the structure of abelian sheaf. This is quite easy to do by hand, and we leave it to the reader to find a good simple argument. The argument we use here, which generalizes to presheaves of algebraic structures is the following: Lemma \ref{lemma-diagram-fibre-product} show that $\mathcal{F}^\#(U)$ is the fibre product of a diagram of abelian groups. Thus $\mathcal{F}^\#$ is an abelian subgroup as desired. \medskip\noindent Note that at this point $\mathcal{F}^\#_x$ is an abelian group by Lemma \ref{lemma-stalk-abelian-presheaf} and that $\mathcal{F}_x \to \mathcal{F}^\#_x$ is a bijection (Lemma \ref{lemma-stalk-sheafification}) and a homomorphism of abelian groups. Hence $\mathcal{F}_x \to \mathcal{F}^\#_x$ is an isomorphism of abelian groups. This will be used below without further mention. \medskip\noindent To prove the adjointness property we use the adjointness property of sheafification of presheaves of sets. For example if $\psi : \mathcal{F} \to i(\mathcal{G})$ is morphism of presheaves then we obtain a morphism of sheaves $\psi' : \mathcal{F}^\# \to \mathcal{G}$. What we have to do is to check that this is a morphism of abelian sheaves. We may do this for example by noting that it is true on stalks, by Lemma \ref{lemma-stalk-sheafification}, and then using Lemma \ref{lemma-check-homomorphism-stalks} above.
The proof is the same as the proof of Lemma \ref{lemma-sheafify-abelian-presheaf}, with repeated application of Lemma \ref{lemma-image-contained-in} (see also Example \ref{example-application-lemma-image-contained-in}). The main idea however, is to define $\mathcal{F}^\#(U)$ as the fibre product in $\mathcal{C}$ of the diagram $$ \xymatrix{  & \Pi(\mathcal{F})(U) \ar[d] \\ \prod_{x \in U} \mathcal{F}_x \ar[r] & \prod_{x \in U} \Pi(\mathcal{F})_x } $$ compare Lemma \ref{lemma-diagram-fibre-product}.
The colimit is over the partially ordered set consisting of open subsets $V \subset Y$ which contain $f(U)$ with ordering by reverse inclusion. This is a directed partially ordered set, since if $V, V'$ are in it then so is $V \cap V'$. Furthermore, if $U_1 \subset U_2$, then every open neighbourhood of $f(U_2)$ is an open neighbourhood of $f(U_1)$. Hence the system defining $f_p\mathcal{G}(U_2)$ is a subsystem of the one defining $f_p\mathcal{G}(U_1)$ and we obtain a restriction map (for example by applying the generalities in Categories, Lemma \ref{categories-lemma-functorial-colimit}). \medskip\noindent Note that the construction of the colimit is clearly functorial in $\mathcal{G}$, and similarly for the restriction mappings. Hence we have defined $f_p$ as a functor. \medskip\noindent A small useful remark is that there exists a canonical map $\mathcal{G}(U) \to f_p\mathcal{G}(f^{-1}(U))$, because the system of open neighbourhoods of $f(f^{-1}(U))$ contains the element $U$. This is compatible with restriction mappings. In other words, there is a canonical map $i_\mathcal{G} : \mathcal{G} \to f_* f_p \mathcal{G}$. \medskip\noindent Let $\mathcal{F}$ be a presheaf of sets on $X$. Suppose that $\psi : f_p\mathcal{G} \to \mathcal{F}$ is a map of presheaves of sets. The corresponding map $\mathcal{G} \to f_*\mathcal{F}$ is the map $f_*\psi \circ i_\mathcal{G} : \mathcal{G} \to f_* f_p \mathcal{G} \to f_* \mathcal{F}$. \medskip\noindent Another small useful remark is that there exists a canonical map $c_\mathcal{F} : f_p f_* \mathcal{F} \to \mathcal{F}$. Namely, let $U \subset X$ open. For every open neighbourhood $V \supset f(U)$ in $Y$ there exists a map $f_*\mathcal{F}(V) = \mathcal{F}(f^{-1}(V))\to \mathcal{F}(U)$, namely the restriction map on $\mathcal{F}$. And this is compatible with the restriction mappings between values of $\mathcal{F}$ on $f^{-1}$ of varying opens containing $f(U)$. Thus we obtain a canonical map $f_p f_* \mathcal{F}(U) \to \mathcal{F}(U)$. Another trivial verification shows that these maps are compatible with restriction maps and define a map $c_\mathcal{F}$ of presheaves of sets. \medskip\noindent Suppose that $\varphi : \mathcal{G} \to f_*\mathcal{F}$ is a map of presheaves of sets. Consider $f_p\varphi : f_p \mathcal{G} \to f_p f_* \mathcal{F}$. Postcomposing with $c_\mathcal{F}$ gives the desired map $c_\mathcal{F} \circ f_p\varphi : f_p\mathcal{G} \to \mathcal{F}$. We omit the verification that this construction is inverse to the construction in the other direction given above.
This you can see as follows \begin{eqnarray*} (f_p\mathcal{G})_x & = & \colim_{x \in U} f_p\mathcal{G}(U) \\ & = & \colim_{x \in U} \colim_{f(U) \subset V} \mathcal{G}(V) \\ & = & \colim_{f(x) \in V} \mathcal{G}(V) \\ & = & \mathcal{G}_{f(x)} \end{eqnarray*} Here we have used Categories, Lemma \ref{categories-lemma-colimits-commute}, and the fact that any $V$ open in $Y$ containing $f(x)$ occurs in the third description above. Details omitted.
Let $U \subset X$ is open. We define the map of the lemma to be the map \begin{eqnarray*} f_p\mathcal{O}(U) \times f_p\mathcal{G}(U) & = & \colim_{f(U) \subset V} \mathcal{O}(V) \times \colim_{f(U) \subset V} \mathcal{G}(V) \\ & = & \colim_{f(U) \subset V} (\mathcal{O}(V)\times \mathcal{G}(V)) \\ & \to & \colim_{f(U) \subset V} \mathcal{G}(V) \\ & = & f_p\mathcal{G}(U). \end{eqnarray*} Here the arrow in the middle is the multiplication map on $Y$. The second equality holds because directed colimits commute with finite limits, see Categories, Lemma \ref{categories-lemma-directed-commutes}. We leave it to the reader to see this is compatible with restriction mappings and defines a structure of $f_p\mathcal{O}$-module on $f_p\mathcal{G}$.
This follows from the equalities \begin{eqnarray*} \Mor_{\textit{PMod}(\mathcal{O})}( \mathcal{O} \otimes_{p, f_pf_*\mathcal{O}} f_p\mathcal{G}, \mathcal{F}) & = & \Mor_{\textit{PMod}(f_pf_*\mathcal{O})}( f_p\mathcal{G}, \mathcal{F}_{f_pf_*\mathcal{O}}) \\ & = & \Mor_{\textit{PMod}(f_*\mathcal{O})}(\mathcal{G}, f_*(\mathcal{F}_{f_pf_*\mathcal{O}})) \\ & = & \Mor_{\textit{PMod}(f_*\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}). \end{eqnarray*} The first equality is Lemma \ref{lemma-adjointness-tensor-restrict-presheaves}. The second equality is Lemma \ref{lemma-adjoint-push-pull-presheaves-modules}. The third equality is given by the equality $f_*(\mathcal{F}_{f_pf_*\mathcal{O}}) = f_*\mathcal{F}$ of abelian sheaves which is $f_*\mathcal{O}$-linear. Namely, $\text{id}_{f_*\mathcal{O}}$ corresponds to $c_\mathcal{O}$ under the adjunction described in the proof of Lemma \ref{lemma-pullback-presheaves} and thus $\text{id}_{f_*\mathcal{O}} = f_*c_\mathcal{O} \circ i_{f_*\mathcal{O}}$.
Argue by the equalities \begin{eqnarray*} \Mor_{\textit{Mod}(f^{-1}\mathcal{O})}(f^{-1}\mathcal{G}, \mathcal{F}) & = & \Mor_{\textit{Mod}(f_p\mathcal{O})}(f_p\mathcal{G}, \mathcal{F}) \\ & = & \Mor_{\textit{Mod}(\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}). \end{eqnarray*} where the second is Lemmas \ref{lemma-adjoint-push-pull-presheaves-modules} and the first is by Lemma \ref{lemma-sheafification-presheaf-modules}.
This follows from the equalities \begin{eqnarray*} \Mor_{\textit{Mod}(\mathcal{O})}( \mathcal{O} \otimes_{f^{-1}f_*\mathcal{O}} f^{-1}\mathcal{G}, \mathcal{F}) & = & \Mor_{\textit{Mod}(f^{-1}f_*\mathcal{O})}( f^{-1}\mathcal{G}, \mathcal{F}_{f^{-1}f_*\mathcal{O}}) \\ & = & \Mor_{\textit{Mod}(f_*\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}). \end{eqnarray*} which are a combination of Lemma \ref{lemma-adjointness-tensor-restrict} and \ref{lemma-adjoint-push-pull-modules}.
This follows from the work we did before: \begin{eqnarray*} \Hom_{\mathcal{O}_X}(f^*\mathcal{G}, \mathcal{F}) & = & \Mor_{\textit{Mod}(\mathcal{O}_X)}( \mathcal{O}_X \otimes_{f^{-1}\mathcal{O}_Y} f^{-1}\mathcal{G}, \mathcal{F}) \\ & = & \Mor_{\textit{Mod}(f^{-1}\mathcal{O}_Y)}( f^{-1}\mathcal{G}, \mathcal{F}_{f^{-1}\mathcal{O}_Y}) \\ & = & \Hom_{\mathcal{O}_Y}(\mathcal{G}, f_*\mathcal{F}). \end{eqnarray*} Here we use Lemmas \ref{lemma-adjointness-tensor-restrict} and \ref{lemma-adjoint-push-pull-modules}.
Assume all the transition maps are injective. In this case the presheaf $\mathcal{F}' : V \mapsto \colim_i \mathcal{F}_i(V)$ is separated (see Definition \ref{definition-separated}). By the discussion above we have $(\mathcal{F}')^\# = \colim_i \mathcal{F}_i$. By Lemma \ref{lemma-separated-presheaf-into-sheaf} we see that $\mathcal{F}' \to (\mathcal{F}')^\#$ is injective. This proves (1). \medskip\noindent Assume $U$ is quasi-compact. Suppose that $s \in \mathcal{F}_i(U)$ and $s' \in \mathcal{F}_{i'}(U)$ give rise to elements on the left hand side which have the same image under $\Psi$. Since $U$ is quasi-compact this means there exists a finite open covering $U = \bigcup_{j = 1, \ldots, m} U_j$ and for each $j$ an index $i_j \in I$, $i_j \geq i$, $i_j \geq i'$ such that $\varphi_{ii_j}(s) = \varphi_{i'i_j}(s')$. Let $i''\in I$ be $\geq$ than all of the $i_j$. We conclude that $\varphi_{ii''}(s)$ and $\varphi_{i'i''}(s)$ agree on the opens $U_j$ for all $j$ and hence that $\varphi_{ii''}(s) = \varphi_{i'i''}(s)$. This proves (2). \medskip\noindent Assume $U$ is quasi-compact and all transition maps injective. Let $s$ be an element of the target of $\Psi$. Since $U$ is quasi-compact there exists a finite open covering $U = \bigcup_{j = 1, \ldots, m} U_j$, for each $j$ an index $i_j \in I$ and $s_j \in \mathcal{F}_{i_j}(U_j)$ such that $s|_{U_j}$ comes from $s_j$ for all $j$. Pick $i \in I$ which is $\geq$ than all of the $i_j$. By (1) the sections $\varphi_{i_ji}(s_j)$ agree over the overlaps $U_j \cap U_{j'}$. Hence they glue to a section $s' \in \mathcal{F}_i(U)$ which maps to $s$ under $\Psi$. This proves (3). \medskip\noindent Assume the hypothesis of (4). In particular we see that $U$ is quasi-compact and hence by (2) we have injectivity of $\Psi$. Let $s$ be an element of the target of $\Psi$. By assumption there  exists a finite open covering $U = \bigcup_{j = 1, \ldots, m} U_j$, with $U_j \cap U_{j'}$ quasi-compact for all $j, j' \in J$ and for each $j$ an index $i_j \in I$ and $s_j \in \mathcal{F}_{i_j}(U_j)$ such that $s|_{U_j}$ is the image of $s_j$ for all $j$. Since $U_j \cap U_{j'}$ is quasi-compact we can apply (2) and we see that there exists an $i_{jj'} \in I$, $i_{jj'} \geq i_j$, $i_{jj'} \geq i_{j'}$ such that $\varphi_{i_ji_{jj'}}(s_j)$ and $\varphi_{i_{j'}i_{jj'}}(s_{j'})$ agree over $U_j \cap U_{j'}$. Choose an index $i \in I$ wich is bigger or equal than all the $i_{jj'}$. Then we see that the sections $\varphi_{i_ji}(s_j)$ of $\mathcal{F}_i$ glue to a section of $\mathcal{F}_i$ over $U$. This section is mapped to the element $s$ as desired.