hcju/stacksqa huggingface dataset:
The assumptions imply that the assumptions of Lemma \ref{lemma-almost-directed} are satisfied (see the discussion in Categories, Section \ref{categories-section-finite-limits}). By Categories, Lemma \ref{categories-lemma-split-into-directed} we see that $\mathcal{I}_V$ is a (possibly empty) disjoint union of directed categories. Hence it suffices to show that $\mathcal{I}_V$ is connected. \medskip\noindent First, we show that $\mathcal{I}_V$ is nonempty. Namely, let $X$ be the final object of $\mathcal{C}$, which exists by assumption. Let $V \to u(X)$ be the morphism coming from the fact that $u(X)$ is final in $\mathcal{D}$ by assumption. This gives an object of $\mathcal{I}_V$. \medskip\noindent Second, we show that $\mathcal{I}_V$ is connected. Let $\phi_1 : V \to u(U_1)$ and $\phi_2 : V \to u(U_2)$ be in $\Ob(\mathcal{I}_V)$. By assumption $U_1\times U_2$ exists and $u(U_1\times U_2) = u(U_1)\times u(U_2)$. Consider the morphism $\phi : V \to u(U_1\times U_2)$ corresponding to $(\phi_1, \phi_2)$ by the universal property of products. Clearly the object $\phi : V \to u(U_1\times U_2)$ maps to both $\phi_1 : V \to u(U_1)$ and $\phi_2 : V \to u(U_2)$.
Let $\mathcal{G}$ be a presheaf on $\mathcal{D}$ and let $\mathcal{F}$ be a presheaf on $\mathcal{C}$. We will show that the displayed formula holds by constructing maps either way. We will leave it to the reader to verify they are each others inverse. \medskip\noindent Given a map $\alpha : u_p \mathcal{F} \to \mathcal{G}$ we get $u^p\alpha : u^p u_p \mathcal{F} \to u^p \mathcal{G}$. Lemma \ref{lemma-recover} says that there is a map $\mathcal{F} \to u^p u_p \mathcal{F}$. The composition of the two gives the desired map. (The good thing about this construction is that it is clearly functorial in everything in sight.) \medskip\noindent Conversely, given a map $\beta : \mathcal{F} \to u^p\mathcal{G}$ we get a map $u_p\beta : u_p\mathcal{F} \to u_p u^p\mathcal{G}$. We claim that the functor $u^p\mathcal{G}_Y$ on $\mathcal{I}_Y$ has a canonical map to the constant functor with value $\mathcal{G}(Y)$. Namely, for every object $(X, \phi)$ of $\mathcal{I}_Y$, the value of $u^p\mathcal{G}_Y$ on this object is $\mathcal{G}(u(X))$ which maps to $\mathcal{G}(Y)$ by $\mathcal{G}(\phi) = \phi^* $. This is a transformation of functors because $\mathcal{G}$ is a functor itself. This leads to a map $u_p u^p \mathcal{G}(Y) \to \mathcal{G}(Y)$. Another trivial verification shows that this is functorial in $Y$ leading to a map of presheaves $u_p u^p \mathcal{G} \to \mathcal{G}$. The composition $u_p\mathcal{F} \to u_p u^p\mathcal{G} \to \mathcal{G}$ is the desired map.
By Lemma \ref{lemma-compare-separated-presheaf-condition} the map $\mathcal{F}(U) \to \prod \mathcal{F}(U_i)$ is injective. Suppose given $s_i \in \mathcal{F}(U_i)$ such that $s_i|_{U_i \times_U U_{i'}} = s_{i'}|_{U_i \times_U U_{i'}}$ for all $i, i' \in I$. Set $s_j = f_j^*(s_{\alpha(j)}) \in \mathcal{F}(V_j)$. Since the morphisms $f_j$ are morphisms over $U$ we obtain induced morphisms $f_{jj'} : V_j \times_U V_{j'} \to U_{\alpha(i)} \times_U U_{\alpha(i')}$ compatible with the $f_j, f_{j'}$ via the projection maps. It follows that $$ s_j|_{V_j \times_U V_{j'}} = f_{jj'}^*(s_{\alpha(j)}|_{U_{\alpha(j)} \times_U U_{\alpha(j')}}) = f_{jj'}^*(s_{\alpha(j')}|_{U_{\alpha(j)} \times_U U_{\alpha(j')}}) = s_{j'}|_{V_j \times_U V_{j'}} $$ for all $j, j' \in J$. Hence, by the sheaf condition for $\mathcal{F}$ with respect to $\mathcal{V}$, we get a section $s \in \mathcal{F}(U)$ which restricts to $s_j$ on each $V_j$. We are done if we show $s$ restricts to $s_i$ on $U_i$ for any $i \in I$. Since $\mathcal{F}$ satisfies (3) it suffices to show that $s$ and $s_i$ restrict to the same element over $U_i \times_U V_j$ for all $j \in J$. To see this we use $$ s|_{U_i \times_U V_j} = s_j|_{U_i \times_U V_j} = (\text{id} \times f_j)^*s_{\alpha(j)}|_{U_i \times_U U_{\alpha(j)}} = (\text{id} \times f_j)^*s_i|_{U_i \times_U U_{\alpha(j)}} = s_i|_{U_i \times_U V_j} $$ as desired.
Part (1) follows directly from Lemma \ref{lemma-tautological-same-sheaf} and the definitions. \medskip\noindent Proof of (2). Let $\mathcal{F}$ be a sheaf of sets for the site $(\mathcal{C}, \text{Cov}_2)$. Let $\mathcal{U} \in \text{Cov}_1$, say $\mathcal{U} = \{U_i \to U\}_{i \in I}$. By assumption we may choose a refinement $\mathcal{V} \in \text{Cov}_2$ of $\mathcal{U}$, say $\mathcal{V} = \{V_j \to U\}_{j \in J}$ and refinement given by $\alpha : J \to I$ and $f_j : V_j \to U_{\alpha(j)}$. Observe that $\mathcal{F}$ satisfies the sheaf condition for $\mathcal{V}$ and for the coverings $\{V_j \times_U U_i \to U_i\}_{j \in J}$ as these are in $\text{Cov}_2$. Hence $\mathcal{F}$ satisfies the sheaf condition for $\mathcal{U}$ by Lemma \ref{lemma-compare-sheaf-condition}.
This is clear from Lemmas \ref{lemma-refine-same-topology} and \ref{lemma-tautological-combinatorial} above as the hypothesis implies that every covering $\mathcal{U} \in \text{Cov}_1 \subset \text{Cov}(\mathcal{C})$ is combinatorially equivalent to an element of $\text{Cov}_2$, and similarly with the roles of $\text{Cov}_1$ and $\text{Cov}_2$ reversed.
Namely, let $\{U_i \to U\}$ be a covering such that $s$ arises from the element $(s_i) \in H^0(\{U_i \to U\}, \mathcal{F})$. According to Lemma \ref{lemma-independent-refinement} we may consider the covering $\{U_i \to U_i\}$ and the (obvious) morphism of coverings $\{U_i \to U_i\} \to \{U_i \to U\}$ to compute the pullback of $s$ to an element of $\mathcal{F}^+(U_i)$. And indeed, using this covering we get exactly $\theta(s_i)$ for the restriction of $s$ to $U_i$.
Since the sheafification functor is a left adjoint it commutes with all colimits, see Categories, Lemma \ref{categories-lemma-adjoint-exact}. Hence, since $\textit{PSh}(\mathcal{C})$ has colimits, we deduce that $\Sh(\mathcal{C})$ has colimits (which are the sheafifications of the colimits in presheaves).
Since it is a left adjoint it is right exact, see Categories, Lemma \ref{categories-lemma-exact-adjoint}. On the other hand, by Lemmas \ref{lemma-common-refinement} and Lemma \ref{lemma-independent-refinement} the colimits in the construction of $\mathcal{F}^+$ are really over the directed set $\Ob(\mathcal{J}_U)$ where $\mathcal{U} \geq \mathcal{U}'$ if and only if $\mathcal{U}$ is a refinement of $\mathcal{U}'$. Hence by Categories, Lemma \ref{categories-lemma-directed-commutes} we see that $\mathcal{F} \to \mathcal{F}^+$ commutes with finite limits (as a functor from presheaves to presheaves). Then we conclude using Lemma \ref{lemma-limit-sheaf}.
Let $\mathcal{H}$ be a sheaf of sets and let $\varphi : \mathcal{F} \to \mathcal{H}$ be a map of sheaves equalizing the two maps $\mathcal{F} \times_\mathcal{G} \mathcal{F} \to \mathcal{F}$. Let $\mathcal{G}' \subset \mathcal{G}$ be the presheaf image of the map $\mathcal{F} \to \mathcal{G}$. As the product $\mathcal{F} \times_\mathcal{G} \mathcal{F}$ may be computed in the category of presheaves we see that it is equal to the presheaf product $\mathcal{F} \times_{\mathcal{G}'} \mathcal{F}$. Hence $\varphi$ induces a unique map of presheaves $\psi' : \mathcal{G}' \to \mathcal{H}$. Since $\mathcal{G}$ is the sheafification of $\mathcal{G}'$ by Lemma \ref{lemma-mono-epi-sheaves} we conclude that $\psi'$ extends uniquely to a map of sheaves $\psi : \mathcal{G} \to \mathcal{H}$. We omit the verification that $\varphi$ is equal to the composition of $\psi$ and the given map.
By Lemma \ref{lemma-mono-epi-sheaves} above we have to show that $\coprod\nolimits_{i \in I} h_{U_i}^\# \to h_U^\#$ is an epimorphism. Let $\mathcal{F}$ be a sheaf of sets. A morphism $h_U^\# \to \mathcal{F}$ corresponds to a section $s \in \mathcal{F}(U)$. Hence the injectivity of $\Mor(h_U^\#, \mathcal{F}) \to \prod_i \Mor(h_{U_i}^\#, \mathcal{F})$ follows directly from the sheaf property of $\mathcal{F}$.
First we show there is an epimorphism $\mathcal{F}_0 \to \mathcal{F}$ of the desired type. Namely, just take $$ \mathcal{F}_0 = \coprod\nolimits_{U \in E, s \in \mathcal{F}(U)} (h_U)^\# \longrightarrow \mathcal{F} $$ Here the arrow restricted to the component corresponding to $(U, s)$ maps the element $\text{id}_U \in h_U^\#(U)$ to the section $s \in \mathcal{F}(U)$. This is an epimorphism according to Lemma \ref{lemma-mono-epi-sheaves} and our condition on $E$. To construct $\mathcal{F}_1$ first set $\mathcal{G} = \mathcal{F}_0 \times_\mathcal{F} \mathcal{F}_0$ and then construct an epimorphism $\mathcal{F}_1 \to \mathcal{G}$ as above. See Lemma \ref{lemma-coequalizer-surjection}.
Since $u_s$ is the left adjoint of $u^s$ we see that $u_s$ is right exact, see Categories, Lemma \ref{categories-lemma-exact-adjoint}. Hence it suffices to show that $u_s$ is left exact. In other words we have to show that $u_s$ commutes with finite limits. Because the categories $\mathcal{I}_Y^{opp}$ are filtered we see that $u_p$ commutes with finite limits, see Categories, Lemma \ref{categories-lemma-directed-commutes} (this also uses the description of limits in $\textit{PSh}$, see Section \ref{section-limits-colimits-PSh}). And since sheafification commutes with finite limits as well (Lemma \ref{lemma-sheafification-exact}) we conclude because $u_s = \# \circ u_p$.
The implications are immediate from the definitions. Let $X = [0, 1] \subset \mathbf{R}$ as a topological space (with the usual $\epsilon$-$\delta$ topology). Let $\mathcal{C}$ be the category of open subspaces of $X$ with inclusions as morphisms and usual open coverings (compare with Example \ref{example-site-topological}). However, then we change the notion of covering in $\mathcal{C}$ to exclude all finite coverings, except for the coverings of the form $\{U \to U\}$. It is easy to see that this will be a site as in Definition \ref{definition-site}. In this site the object $X = U = [0, 1]$ is quasi-compact in the sense of Definition \ref{definition-quasi-compact} but $U$ does not satisfy (2). We leave it to the reader to make an example where (2) holds but not (3).
Assume (1) and let $\coprod_{i \in I} \mathcal{F}_i \to h_U^\#$ be a surjection. Then $\text{id}_U$ is a section of $h_U^\#$ over $U$. Hence there exists a covering $\{U_a \to U\}_{a \in A}$ and for each $a \in A$ a section $s_a$ of $\coprod_{i \in I} \mathcal{F}_i$ over $U_a$ mapping to $\text{id}_U$. By the construction of coproducts as sheafification of coproducts of presheaves (Lemma \ref{lemma-colimit-sheaves}), for each $a$ there exists a covering $\{U_{ab} \to U_a\}_{b \in B_a}$ and for all $b \in B_a$ an $\iota(b) \in I$ and a section $s_{b}$ of $\mathcal{F}_{\iota(b)}$ over $U_{ab}$ mapping to $\text{id}_U|_{U_{ab}}$. Thus after replacing the covering $\{U_a \to U\}_{a \in A}$ by $\{U_{ab} \to U\}_{a \in A, b \in B_a}$ we may assume we have a map $\iota : A \to I$ and for each $a \in A$ a section $s_a$ of $\mathcal{F}_{\iota(a)}$ over $U_a$ mapping to $\text{id}_U$. Since $U$ is quasi-compact, there is a covering $\{V_c \to U\}_{c \in C}$, a map $\alpha : C \to A$ with finite image, and $V_c \to U_{\alpha(c)}$ over $U$. Then we see that $J = \Im(\iota \circ \alpha) \subset I$ works because $\coprod_{c \in C} h_{V_c}^\# \to h_U^\#$ is surjective (Lemma \ref{lemma-covering-surjective-after-sheafification}) and factors through $\coprod_{i \in J} \mathcal{F}_i \to h_U^\#$. (Here we use that the composition $h_{V_c}^\# \to h_{U_{\alpha(c)}} \xrightarrow{s_{\alpha(c)}} \mathcal{F}_{\iota(\alpha(c))} \to h_U^\#$ is the map $h_{V_c}^\# \to h_U^\#$ coming from the morphism $V_c \to U$ because $s_{\alpha(c)}$ maps to $\text{id}_U|_{U_{\alpha(c)}}$.) \medskip\noindent Assume (2). Let $\{U_i \to U\}_{i \in I}$ be a covering. By Lemma \ref{lemma-covering-surjective-after-sheafification} we see that $\coprod_{i \in I} h_{U_i}^\# \to h_U^\#$ is surjective. Thus we find a finite subset $J \subset I$ such that $\coprod_{j \in J} h_{U_j}^\# \to h_U^\#$ is surjective. Then arguing as above we find a covering $\{V_c \to U\}_{c \in C}$ of $U$ in $\mathcal{C}$ and a map $\iota : C \to J$ such that $\text{id}_U$ lifts to a section of $s_c$ of $h_{U_{\iota(c)}}^\#$ over $V_c$. Refining the covering even further we may assume $s_c \in h_{U_{\iota(c)}}(V_c)$ mapping to $\text{id}_U$. Then $s_c : V_c \to U_{\iota(c)}$ is a morphism over $U$ and we conclude.
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 Lemma \ref{lemma-colimit-sheaves} we have $(\mathcal{F}')^\# = \colim_i \mathcal{F}_i$. By Theorem \ref{theorem-plus} 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 $\Psi$. This means we can choose a covering $\{U_a \to U\}_{a \in A}$ and for each $a \in A$ an index $i_a \in I$, $i_a \geq i$, $i_a \geq i'$ such that $\varphi_{ii_a}(s) = \varphi_{i'i_a}(s')$. Because $U$ is quasi-compact we can choose a covering $\{V_b \to U\}_{b \in B}$, a map $\alpha : B \to A$ with finite image, and morphisms $V_b \to U_{\alpha(b)}$ over $U$. Pick $i''\in I$ to be $\geq$ than all of the $i_{\alpha(b)}$ which is possible because the image of $\alpha$ is finite. We conclude that $\varphi_{ii''}(s)$ and $\varphi_{i'i''}(s)$ agree on $V_b$ for all $b \in B$ 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 $\Psi$. There exists a covering $\{U_a \to U\}_{a \in A}$ and for each $a \in A$ an index $i_a \in I$ and a section $s_a \in \mathcal{F}_{i_a}(U_a)$ such that $s|_{U_a}$ comes from $s_a$ for all $a \in A$. Because $U$ is quasi-compact we can choose a covering $\{V_b \to U\}_{b \in B}$, a map $\alpha : B \to A$ with finite image, and morphisms $V_b \to U_{\alpha(b)}$ over $U$. Pick $i \in I$ to be $\geq$ than all of the $i_{\alpha(b)}$ which is possible because the image of $\alpha$ is finite. By (1) the sections $s_b = \varphi_{i_{\alpha(b)} i}(s_{\alpha(b)})|_{V_b}$ agree over $V_b \times_U V_{b'}$. 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). Let $s$ be an element of the target of $\Psi$. By assumption there  exists a finite covering $\{U_j \to U\}_{j = 1, \ldots, m} U_j$, with $U_j \times_U U_{j'}$ 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 $U_j \times_U U_{j'}$ 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 $U_j \times_U U_{j'}$. 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.
Our definition of composition of morphisms of sites implies that $u_b \circ u_a = u_c$ whenever $c = a \circ b$ in $\mathcal{I}$. The formula $\mathcal{C} = \colim \mathcal{C}_i$ means that $\Ob(\mathcal{C}) = \colim \Ob(\mathcal{C}_i)$ and $\text{Arrows}(\mathcal{C}) = \colim \text{Arrows}(\mathcal{C}_i)$. Then source, target, and composition are inherited from the source, target, and composition on $\text{Arrows}(\mathcal{C}_i)$. In this way we obtain a category.  Denote $u_i : \mathcal{C}_i \to \mathcal{C}$ the obvious functor. Remark that given any finite diagram in $\mathcal{C}$ there exists an $i$ such that this diagram is the image of a diagram in $\mathcal{C}_i$. \medskip\noindent Let $\{U^t \to U\}$ be a covering of $\mathcal{C}$. We first prove that if $V \to U$ is a morphism of $\mathcal{C}$, then $U^t \times_U V$ exists. By our remark above and our definition of coverings, we can find an $i$, a covering $\{U_i^t \to U_i\}$ of $\mathcal{C}_i$ and a morphism $V_i \to U_i$ whose image by $u_i$ is the given data. We claim that $U^t \times_U V$ is the image of $U^t_i \times_{U_i} V_i$ by $u_i$. Namely, for every $a : j \to i$ in $\mathcal{I}$ the functor $u_a$ is continuous, hence $u_a(U^t_i \times_{U_i} V_i) = u_a(U^t_i) \times_{u_a(U_i)} u_a(V_i)$. In particular we can replace $i$ by $j$, if we so desire. Thus, if $W$ is another object of $\mathcal{C}$, then we may assume $W = u_i(W_i)$ and we see that \begin{align*} & \Mor_\mathcal{C}(W, u_i(U^t_i \times_{U_i} V_i)) \\ & = \colim_{a : j \to i} \Mor_{\mathcal{C}_j}(u_a(W_i), u_a(U^t_i \times_{U_i} V_i)) \\ & = \colim_{a : j \to i} \Mor_{\mathcal{C}_j}(u_a(W_i), u_a(U^t_i)) \times_{\Mor_{\mathcal{C}_j}(u_a(W_i), u_a(U_i))} \Mor_{\mathcal{C}_j}(u_a(W_i), u_a(V_i)) \\ & = \Mor_\mathcal{C}(W, U^t) \times_{\Mor_\mathcal{C}(W, U)} \Mor_\mathcal{C}(W, V) \end{align*} as filtered colimits commute with finite limits (Categories, Lemma \ref{categories-lemma-directed-commutes}). It also follows that $\{U^t \times_U V \to V\}$ is a covering in $\mathcal{C}$. In this way we see that axiom (3) of Definition \ref{definition-site} holds. \medskip\noindent To verify axiom (2) of Definition \ref{definition-site} let $\{U^t \to U\}_{t \in T}$ be a covering of $\mathcal{C}$ and for each $t$ let $\{U^{ts} \to U^t\}$ be a covering of $\mathcal{C}$. Then we can find an $i$ and a covering $\{U^t_i \to U_i\}_{t \in T}$ of $\mathcal{C}_i$ whose image by $u_i$ is $\{U^t \to U\}$. Since $T$ is {\bf finite} we may choose an $a : j \to i$ in $\mathcal{I}$ and coverings $\{U^{ts}_j \to u_a(U^t_i)\}$ of $\mathcal{C}_j$ whose image by $u_j$ gives $\{U^{ts} \to U^t\}$. Then we conclude that $\{U^{ts} \to U\}$ is a covering of $\mathcal{C}$ by an application of axiom (2) to the site $\mathcal{C}_j$. \medskip\noindent We omit the proof of axiom (1) of Definition \ref{definition-site}.
A formal argument shows that $$ \colim_{a : j \to i} \mathcal{F}_i(u_a(X_i)) = \colim_{a : j \to i} \colim_{b : k \to j} f_b^{-1}\mathcal{F}_j(u_{a \circ b}(X_i)) $$ By (\ref{equation-compute-pullback-to-limit}) we see that the inner colimit is equal to $f_j^{-1}\mathcal{F}_j(u_i(X_i))$ hence we conclude by Lemma \ref{lemma-directed-colimits-sections}.
For the morphism $$ \varphi_a : f_a^{-1}f_{i, *}\mathcal{F} \to f_{j, *}\mathcal{F} $$ we choose the adjoint to the identity map $$ f_{i, *}\mathcal{F} \to f_{a, *}f_{j, *}\mathcal{F} $$ Hence $\varphi_a$ is the counit for the adjunction given by $(f_a^{-1}, f_{a, *})$. We must prove that for all $a : j \to i$ and $b : k \to i$ with composition $c = a \circ b$ we have $\varphi_c = \varphi_b \circ f_b^{-1}\varphi_a$. This follows from Categories, Lemma \ref{categories-lemma-compose-counits}. Lastly, we must prove that the map given by adjunction $$ \colim_{i \in I} f_i^{-1}f_{i, *}\mathcal{F} \longrightarrow \mathcal{F} $$ is an isomorphism. For an object $U$ of $\mathcal{C}$ we need to show the map $$ (\colim_{i \in I} f_i^{-1}\mathcal{F}_i)(U) \to \mathcal{F}(U) $$ is bijective. Choose an $i$ and an object $U_i$ of $\mathcal{C}_i$ with $u_i(U_i) = U$. Then the left hand side is equal to $$ (\colim_{i \in I} f_i^{-1}\mathcal{F}_i)(U) = \colim_{a : j \to i} f_{j, *}\mathcal{F}(u_a(U_i)) $$ by Lemma \ref{lemma-colimit}. Since $u_j(u_a(U_i)) = U$ we have $f_{j, *}\mathcal{F}(u_a(U_i)) = \mathcal{F}(U)$ for all $a : j \to i$ by definition. Hence the value of the colimit is $\mathcal{F}(U)$ and the proof is complete.
This is proved in exactly the same way as the proof of Lemma \ref{lemma-adjoints-u}. We note that the map $u^p{}_pu \mathcal{F} \to \mathcal{F}$ from Lemma \ref{lemma-recover-pu} is the map that is used to go from the right to the left. \medskip\noindent Alternately, think of a presheaf of sets $\mathcal{F}$ on $\mathcal{C}$ as a presheaf $\mathcal{F}'$ on $\mathcal{C}^{opp}$ with values in $\textit{Sets}^{opp}$, and similarly on $\mathcal{D}$. Check that $({}_pu \mathcal{F})' = u_p(\mathcal{F}')$, and that $(u^p\mathcal{G})' = u^p(\mathcal{G}')$. By Remark \ref{remark-functoriality-presheaves-values} we have the adjointness of $u_p$ and $u^p$ for presheaves with values in $\textit{Sets}^{opp}$. The result then follows formally from this.
Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous functor of sites. Let $w : \mathcal{D} \to \mathcal{C}$ be a continuous left adjoint. Then $u_p = w^p$ by Lemma \ref{lemma-adjoint-functors}. Hence $u_s = w^s$ has a left adjoint, namely $w_s$ (Lemma \ref{lemma-adjoint-sheaves}). Thus $u_s$ has both a right and a left adjoint, whence is exact (Categories, Lemma \ref{categories-lemma-exact-adjoint}).
Let $\{V_j \to V\}_{j \in J}$ be a covering of the site $\mathcal{D}$. We have to show that $$ \xymatrix{ &\ \phantom{}_pu\mathcal{F}(V) \ar[r] & \prod {}_pu\mathcal{F}(V_j) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod {}_pu\mathcal{F}(V_j \times_V V_{j'}) & } $$ is an equalizer diagram. Since ${}_pu$ is right adjoint to $u^p$ we have $$ {}_pu\mathcal{F}(V) = \Mor_{\textit{PSh}(\mathcal{D})}(h_V, {}_pu\mathcal{F}) = \Mor_{\textit{PSh}(\mathcal{C})}(u^ph_V, \mathcal{F}) = \Mor_{\Sh(\mathcal{C})}((u^ph_V)^\#, \mathcal{F}) $$ Hence it suffices to show that \begin{equation} \label{equation-coequalizer} \xymatrix{ \coprod u^p h_{V_j \times_V V_{j'}} \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod u^p h_{V_j} \ar[r] & u^p h_V } \end{equation} becomes a coequalizer diagram after sheafification. (Recall that a coproduct in the category of sheaves is the sheafification of the coproduct in the category of presheaves, see Lemma \ref{lemma-colimit-sheaves}.) \medskip\noindent We first show that the second arrow of (\ref{equation-coequalizer}) becomes surjective after sheafification. To do this we use Lemma \ref{lemma-mono-epi-sheaves}. Thus it suffices to show a section $s$ of $u^ph_V$ over $U$ lifts to a section of $\coprod u^p h_{V_j}$ on the members of a covering of $U$. Note that $s$ is a morphism $s : u(U) \to V$. Then $\{V_j \times_{V, s} u(U) \to u(U)\}$ is a covering of $\mathcal{D}$. Hence, as $u$ is cocontinuous, there is a covering $\{U_i \to U\}$ such that $\{u(U_i) \to u(U)\}$ refines $\{V_j \times_{V, s} u(U) \to u(U)\}$. This means that each restriction $s|_{U_i} : u(U_i) \to V$ factors through a morphism $s_i : u(U_i) \to V_j$ for some $j$, i.e., $s|_{U_i}$ is in the image of $u^ph_{V_j}(U_i) \to u^ph_V(U_i)$ as desired. \medskip\noindent Let $s, s' \in (\coprod u^ph_{V_j})^\#(U)$ map to the same element of $(u^ph_V)^\#(U)$. To finish the proof of the lemma we show that after replacing $U$ by the members of a covering that $s, s'$ are the image of the same section of $\coprod u^p h_{V_j \times_V V_{j'}}$ by the two maps of (\ref{equation-coequalizer}). We may first replace $U$ by the members of a covering and assume that $s \in u^ph_{V_j}(U)$ and $s' \in u^ph_{V_{j'}}(U)$. A second such replacement guarantees that $s$ and $s'$ have the same image in $u^ph_V(U)$ instead of in the sheafification. Hence $s : u(U) \to V_j$ and $s' : u(U) \to V_{j'}$ are morphisms of $\mathcal{D}$ such that $$ \xymatrix{ u(U) \ar[r]_{s'} \ar[d]_s & V_{j'} \ar[d] \\ V_j \ar[r] & V } $$ is commutative. Thus we obtain $t = (s, s') : u(U) \to V_j \times_V V_{j'}$, i.e., a section $t \in u^ph_{V_j \times_V V_{j'}}(U)$ which maps to $s, s'$ as desired.
Let us prove the adjointness property as follows \begin{eqnarray*} \Mor_{\Sh(\mathcal{C})} ((u^p\mathcal{G})^\#, \mathcal{F}) & = & \Mor_{\textit{PSh}(\mathcal{C})} (u^p\mathcal{G}, \mathcal{F}) \\ & = & \Mor_{\textit{PSh}(\mathcal{D})} (\mathcal{G}, {}_pu\mathcal{F}) \\ & = & \Mor_{\Sh(\mathcal{D})} (\mathcal{G}, {}_su\mathcal{F}). \end{eqnarray*} Thus it is a left adjoint and hence right exact, see Categories, Lemma \ref{categories-lemma-exact-adjoint}. We have seen that sheafification is left exact, see Lemma \ref{lemma-sheafification-exact}. Moreover, the inclusion $i : \Sh(\mathcal{D}) \to \textit{PSh}(\mathcal{D})$ is left exact by Lemma \ref{lemma-limit-sheaf}. Finally, the functor $u^p$ is left exact because it is a right adjoint (namely to $u_p$). Thus the functor is the composition ${}^\# \circ u^p \circ i$ of left exact functors, hence left exact.