hcju/stacksqa huggingface dataset:
Please note the analogy with Lemma \ref{lemma-directed}. The assumptions on $\mathcal{C}$ imply that $\mathcal{C}$ has finite limits. See Categories, Lemma \ref{categories-lemma-finite-limits-exist}. Assumption (1) implies that the category of neighbourhoods is nonempty. Suppose $(U, x)$ and $(V, y)$ are neighbourhoods. Then $u(U \times V) = u(U \times_X V) = u(U) \times_{u(X)} u(V) = u(U) \times u(V)$ by (2). Hence there exists a neighbourhood $(U \times_X V, z)$ mapping to both $(U, x)$ and $(V, y)$. Let $a, b : (V, y) \to (U, x)$ be two morphisms in the category of neighbourhoods. Let $W$ be the equalizer of $a, b : V \to U$. As in the proof of Categories, Lemma \ref{categories-lemma-finite-limits-exist} we may write $W$ in terms of fibre products: $$ W = (V \times_{a, U, b} V) \times_{(pr_1, pr_2), V \times V, \Delta} V $$ The bijectivity in (2) guarantees there exists an element $z \in u(W)$ which maps to $((y, y), y)$. Then $(W, z) \to (V, y)$ equalizes $a, b$ as desired. The ``consequently'' clause is Lemma \ref{lemma-neighbourhoods-cofiltered}.
This is a simple categorical fact. We have \begin{align*} (u_p\mathcal{F})_q & = \colim_{(V, y)} \colim_{U, \phi : V \to u(U)} \mathcal{F}(U) \\ & = \colim_{(V, y, U, \phi : V \to u(U))} \mathcal{F}(U) \\ & = \colim_{(U, x)} \mathcal{F}(U) \\ & = \mathcal{F}_p \end{align*} The first equality holds by the definition of $u_p$ and the definition of the stalk functor. Observe that $y \in v(V)$. In the second equality we simply combine colimits. To see the third equality we apply Categories, Lemma \ref{categories-lemma-colimit-constant-connected-fibers} to the functor $F$ of diagram categories defined by the rule $$ F((V, y, U, \phi : V \to u(U))) = (U, v(\phi)(y)). $$ This makes sense because $w(U) = v(u(U))$. Let us check the hypotheses of Categories, Lemma \ref{categories-lemma-colimit-constant-connected-fibers}. Observe that $F$ has a right inverse, namely $(U, x) \mapsto (u(U), x, U, \text{id} : u(U) \to u(U))$. Again this makes sense because $x \in w(U) = v(u(U))$. On the other hand, there is always a morphism $$ (V, y, U, \phi : V \to u(U)) \longrightarrow (u(U), v(\phi)(y), U, \text{id} : u(U) \to u(U)) $$ in the fibre category over $(U, x)$ which shows the fibre categories are connected. The fourth and final equality is clear.
[First proof Lemma \ref{lemma-point-morphism-sites}] Note that since $u$ is continuous and since $v$ defines a point, it is immediate that $v \circ u$ satisfies conditions (1) and (2) of Definition \ref{definition-point}. Let us prove the displayed equality. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Then $$ (f^{-1}\mathcal{F})_q = (u_s\mathcal{F})_q = (u_p \mathcal{F})_q = \mathcal{F}_p $$ The first equality since $f^{-1} = u_s$, the second equality by Lemma \ref{lemma-point-pushforward-sheaf}, and the third by Lemma \ref{lemma-point-functor}. Hence now we see that $p$ also satisfies condition (3) of Definition \ref{definition-point} because it is a composition of exact functors. This finishes the proof.
Choose $S$ and $\mathcal{S}$ as in Lemma \ref{lemma-site-point-morphism}. We may identify $\Sh(pt) = \Sh(\mathcal{S})$ as in that lemma, and we may write $p = f : \Sh(\mathcal{S}) \to \Sh(\mathcal{C})$ for the morphism of topoi induced by $u$. By Lemma \ref{lemma-localize-morphism} we get a commutative diagram of topoi $$ \xymatrix{ \Sh(\mathcal{S}/u(U)) \ar[r]_-{j_{u(U)}} \ar[d]_{p'} & \Sh(\mathcal{S}) \ar[d]^p \\ \Sh(\mathcal{C}/U) \ar[r]^{j_U} & \Sh(\mathcal{C}), } $$ where $p'$ is given by the functor $u' : \mathcal{C}/U \to \mathcal{S}/u(U)$, $V/U \mapsto u(V)/u(U)$. Consider the functor $j_x : \mathcal{S} \cong \mathcal{S}/x$ obtained by assigning to a set $E$ the set $E$ endowed with the constant map $E \to u(U)$ with value $x$. Then $j_x$ is a fully faithful cocontinuous functor which has a continuous right adjoint $v_x : (\psi : E \to u(U)) \mapsto \psi^{-1}(\{x\})$. Note that $j_{u(U)} \circ j_x = \text{id}_\mathcal{S}$, and $v_x \circ u' = v$. These observations imply that we have the following commutative diagram of topoi $$ \xymatrix{ \Sh(\mathcal{S}) \ar[rd]^a \ar[rdd]_q \ar `r[rrr] `d[dd]^p [rrdd] & & & \\ & \Sh(\mathcal{S}/u(U)) \ar[r]_-{j_{u(U)}} \ar[d]^{p'} & \Sh(\mathcal{S}) \ar[d]^p & \\ & \Sh(\mathcal{C}/U) \ar[r]^{j_U} & \Sh(\mathcal{C}) & } $$ Namely: \begin{enumerate} \item The morphism $a : \Sh(\mathcal{S}) \to \Sh(\mathcal{S}/u(U))$ is the morphism of topoi associated to the cocontinuous functor $j_x$, which equals the morphism associated to the continuous functor $v_x$, see Lemma \ref{lemma-cocontinuous-morphism-topoi} and Section \ref{section-cocontinuous-adjoint}. \item The composition $p \circ j_{u(U)} \circ a = p$ since $j_{u(U)} \circ j_x = \text{id}_\mathcal{S}$. \item The composition $p' \circ a$ gives a morphism of topoi. Moreover, it is the morphism of topoi associated to the continuous functor $v_x \circ u' = v$. Hence $v$ does indeed define a point $q$ of $\mathcal{C}/U$ which fits into the diagram above by construction. \end{enumerate} This ends the proof of the lemma.
Let $q$ be a point of $\mathcal{C}/U$ given by the functor $v : \mathcal{C}/U \to \textit{Sets}$ such that $j_U \circ q = p$ as morphisms of topoi. Recall that $u(V) = p^{-1}(h_V^\#)$ for any object $V$ of $\mathcal{C}$, see Lemma \ref{lemma-point-site-topos}. Similarly $v(V/U) = q^{-1}(h_{V/U}^\#)$ for any object $V/U$ of $\mathcal{C}/U$. Consider the following two diagrams $$ \vcenter{ \xymatrix{ \Mor_{\mathcal{C}/U}(W/U, V/U) \ar[r] \ar[d] & \Mor_\mathcal{C}(W, V) \ar[d] \\ \Mor_{\mathcal{C}/U}(W/U, U/U) \ar[r] & \Mor_\mathcal{C}(W, U) } } \quad \vcenter{ \xymatrix{ h_{V/U}^\# \ar[r] \ar[d] & j_U^{-1}(h_V^\#) \ar[d] \\ h_{U/U}^\# \ar[r] & j_U^{-1}(h_U^\#) } } $$ The right hand diagram is the sheafification of the diagram of presheaves on $\mathcal{C}/U$ which maps $W/U$ to the left hand diagram of sets. (There is a small technical point to make here, namely, that we have $(j_U^{-1}h_V)^\# = j_U^{-1}(h_V^\#)$ and similarly for $h_U$, see Lemma \ref{lemma-technical-pu}.) Note that the left hand diagram of sets is cartesian. Since sheafification is exact (Lemma \ref{lemma-sheafification-exact}) we conclude that the right hand diagram is cartesian. \medskip\noindent Apply the exact functor $q^{-1}$ to the right hand diagram to get a cartesian diagram $$ \xymatrix{ v(V/U) \ar[r] \ar[d] & u(V) \ar[d] \\ v(U/U) \ar[r] & u(U) } $$ of sets. Here we have used that $q^{-1} \circ j^{-1} = p^{-1}$. Since $U/U$ is a final object of $\mathcal{C}/U$ we see that $v(U/U)$ is a singleton. Hence the image of $v(U/U)$ in $u(U)$ is an element $x$, and the top horizontal map gives a bijection $v(V/U) \to  \{y \in u(V) \mid y \mapsto x\text{ in }u(U)\}$ as desired.
We use the description of $j_{U!}\mathcal{G}$ as the sheaf associated to the presheaf $V \mapsto \coprod\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}(V/_\varphi U)$ of Lemma \ref{lemma-describe-j-shriek}. Also, the stalk of $j_{U!}\mathcal{G}$ at $p$ is equal to the stalk of this presheaf, see Lemma \ref{lemma-point-pushforward-sheaf}. Hence we see that $$ (j_{U!}\mathcal{G})_p = \colim_{(V, y)} \coprod\nolimits_{\varphi : V \to U} \mathcal{G}(V/_\varphi U) $$ To each element $(V, y, \varphi, s)$ of this colimit, we can assign $x = u(\varphi)(y) \in u(U)$. Hence we obtain $$ (j_{U!}\mathcal{G})_p = \coprod\nolimits_{x \in u(U)} \colim_{(\varphi : V \to U, y), \ u(\varphi)(y) = x} \mathcal{G}(V/_\varphi U). $$ This is equal to the expression of the lemma by our construction of the points $q$.
Suppose that the family is conservative and that $\mathcal{F}$, $U$, and $s, s'$ are as in the lemma. The sections $s$, $s'$ define maps $a, a' : (h_U)^\# \to \mathcal{F}$ which are distinct. Hence, by Lemma \ref{lemma-exactness-stalks} there is an $i$ such that $a_{p_i} \not = a'_{p_i}$. Recall that $(h_U)^\#_{p_i} = u_i(U)$, by Lemmas \ref{lemma-points-recover} and \ref{lemma-point-pushforward-sheaf}. Hence there exists an $x \in u_i(U)$ such that $a_{p_i}(x) \not = a'_{p_i}(x)$ in $\mathcal{F}_{p_i}$. Unwinding the definitions you see that $(U, x, s)$ and $(U, x, s')$ are as in the statement of the lemma. \medskip\noindent To prove the converse, assume the condition on the existence of points of the lemma. Let $\phi : \mathcal{F} \to \mathcal{G}$ be a map of sheaves which is an isomorphism at all the stalks. We have to show that $\phi$ is both injective and surjective, see Lemma \ref{lemma-mono-epi-sheaves}. Injectivity is an immediate consequence of the assumption. To show surjectivity we have to show that $\mathcal{G} \amalg_\mathcal{F} \mathcal{G} \to \mathcal{G}$ is an isomorphism (Categories, Lemma \ref{categories-lemma-characterize-mono-epi}). Since this map is clearly surjective, it suffices to check injectivity which follows as $\mathcal{G} \amalg_\mathcal{F} \mathcal{G} \to \mathcal{G}$ is injective on all stalks by assumption.
We know that $j_{U!}$ induces an equivalence $j_{U!} : \Sh(\mathcal{C}/U) \to \Sh(\mathcal{C})/h_U^\#$, see Lemma \ref{lemma-essential-image-j-shriek}. Moreover, we know that $(j_{U!}\mathcal{G})_{p_i} = \coprod_x \mathcal{G}_{q_{i, x}}$, see Lemma \ref{lemma-stalk-j-shriek}. Hence the result follows formally.
For each $i \in I$ let $\{p_j\}_{j \in J_i}$ be a conservative family of points of $\mathcal{C}/U_i$. For $j \in J_i$ denote $q_j : \Sh(pt) \to \Sh(\mathcal{C})$ the composition of $p_j$ with the localization morphism $\Sh(\mathcal{C}/U_i) \to \Sh(\mathcal{C})$. Then $q_j$ is a point, see Lemma \ref{lemma-point-morphism-topoi}. We claim that the family of points $\{q_j\}_{j \in \coprod J_i}$ is conservative. Namely, let $\mathcal{F} \to \mathcal{G}$ be a map of sheaves on $\mathcal{C}$ such that $\mathcal{F}_{q_j} \to \mathcal{G}_{q_j}$ is an isomorphism for all $j \in \coprod J_i$. Let $W$ be an object of $\mathcal{C}$. By assumption (1) there exists a covering $\{W_a \to W\}$ and morphisms $W_a \to U_{i(a)}$. Since $(\mathcal{F}|_{\mathcal{C}/U_{i(a)}})_{p_j} = \mathcal{F}_{q_j}$ and $(\mathcal{G}|_{\mathcal{C}/U_{i(a)}})_{p_j} = \mathcal{G}_{q_j}$ by Lemma \ref{lemma-point-morphism-topoi} we see that $\mathcal{F}|_{U_{i(a)}} \to \mathcal{G}|_{U_{i(a)}}$ is an isomorphism since the family of points $\{p_j\}_{j \in J_{i(a)}}$ is conservative. Hence $\mathcal{F}(W_a) \to \mathcal{G}(W_a)$ is bijective for each $a$. Similarly $\mathcal{F}(W_a \times_W W_b) \to \mathcal{G}(W_a \times_W W_b)$ is bijective for each $a, b$. By the sheaf condition this shows that $\mathcal{F}(W) \to \mathcal{G}(W)$ is bijective, i.e., $\mathcal{F} \to \mathcal{G}$ is an isomorphism.
Denote $p_i$ the stalk functor (\ref{equation-stalk}) on $\textit{PSh}(\mathcal{C})$ corresponding to the functor $u_i$. We have $$ (f^{-1}\mathcal{F})_{q_i} = (u_s\mathcal{F})_{q_i} = (u_p\mathcal{F})_{q_i} = \mathcal{F}_{p_i} $$ The first equality since $f^{-1} = u_s$, the second equality by Lemma \ref{lemma-point-pushforward-sheaf}, and the third by Lemma \ref{lemma-point-functor}. Hence if $p_i$ is a point, then pulling back by $f$ and then taking stalks at $q_i$ is an exact functor. Since the family of points $\{q_i\}$ is conservative, this implies that $f^{-1}$ is an exact functor and we see that $f$ is a morphism of sites by Definition \ref{definition-morphism-sites}.
There exists a $j_0 \in J$ such that $f$ is defined by $f' : V_{j_0} \to W$. For $j \geq j_0$ we set $V_{j, k} = V_j \times_{f'\circ f_{j j_0}, W} W_k$. Then $\{V_{j, k} \to V_j\}$ is a finite covering in the site $\mathcal{C}$. Hence $\mathcal{F}(V_j) \subset \prod_k \mathcal{F}(V_{j, k})$. By Categories, Lemma \ref{categories-lemma-directed-commutes} once again we see that $$ \mathcal{F}_{p'} = \colim_j \mathcal{F}(V_j) \longrightarrow \prod\nolimits_k \colim_j \mathcal{F}(V_{j, k}) $$ is injective. Hence there exists a $k$ such that $s$ and $s'$ have distinct image in $\colim_j \mathcal{F}(V_{j, k})$. Let $J_0 = \{j \in J, j \geq j_0\}$ and $I = J \amalg J_0$. We order $I$ so that no element of the second summand is smaller than any element of the first, but otherwise using the ordering on $J$. If $j \in I$ is in the first summand then we use $V_j$ and if $j \in I$ is in the second summand then we use $V_{j, k}$. We omit the definition of the transition maps of the inverse system. By the above it follows that $s, s'$ have distinct image in $\mathcal{F}_p$. Moreover, the restriction of $f'$ to $V_{j, k}$ factors through $W_k$ by construction.
Let $E$ be the set of pairs $(\{W_k \to W\}, f\in u'(W))$. Consider pairs $(E' \subset E, (I, \geq, U_i, f_{ii'}))$ such that \begin{enumerate} \item $(I, \geq, U_i, g_{ii'})$ is a refinement of $(J, \geq, V_j, g_{jj'})$, \item $s, s'$ map to distinct elements of $\mathcal{F}_p$, and \item for every pair $(\{W_k \to W\}, f\in u'(W)) \in E'$ we have that the image of $f$ in $u(W)$ is in the image of one of the $u(W_k)$. \end{enumerate} We order such pairs by inclusion in the first factor and by refinement in the second. Denote $\mathcal{S}$ the class of all pairs $(E' \subset E, (I, \geq, U_i, f_{ii'}))$ as above. We claim that the hypothesis of Zorn's lemma holds for $\mathcal{S}$. Namely, suppose that $(E'_a, (I_a, \geq, U_i, f_{ii'}))_{a \in A}$ is a totally ordered subset of $\mathcal{S}$. Then we can define $E' = \bigcup_{a \in A} E'_a$ and we can set $I = \bigcup_{a \in A} I_a$. We claim that the corresponding pair $(E' , (I, \geq, U_i, f_{ii'}))$ is an element of $\mathcal{S}$. Conditions (1) and (3) are clear. For condition (2) you note that $$ u = \colim_{a \in A} u_a \text{ and correspondingly } \mathcal{F}_p = \colim_{a \in A} \mathcal{F}_{p_a} $$ The distinctness of the images of $s, s'$ in this stalk follows from the description of a directed colimit of sets, see Categories, Section \ref{categories-section-directed-colimits}. We will simply write $(E', (I, \ldots)) = \bigcup_{a \in A}(E'_a, (I_a, \ldots))$ in this situation. \medskip\noindent OK, so Zorn's Lemma would apply if $\mathcal{S}$ was a set, and this would, combined with Lemma \ref{lemma-refine} above easily prove the lemma. It doesn't since $\mathcal{S}$ is a class. In order to circumvent this we choose a well ordering on $E$. For $e \in E$ set $E'_e = \{e' \in E \mid e' \leq e\}$. By transfinite induction we construct pairs $(E'_e, (I_e, \ldots)) \in \mathcal{S}$ such that $e_1 \leq e_2 \Rightarrow (E'_{e_1}, (I_{e_1}, \ldots)) \leq (E'_{e_2}, (I_{e_2}, \ldots))$. Let $e \in E$, say $e = (\{W_k \to W\}, f\in u'(W))$. If $e$ has a predecessor $e - 1$, then we let $(I_e, \ldots)$ be a refinement of $(I_{e - 1}, \ldots)$ as in Lemma \ref{lemma-refine} with respect to the system $e = (\{W_k \to W\}, f\in u'(W))$. If $e$ does not have a predecessor, then we let $(I_e, \ldots)$ be a refinement of $\bigcup_{e' < e} (I_{e'}, \ldots)$ with respect to the system $e = (\{W_k \to W\}, f\in u'(W))$. Finally, the union $\bigcup_{e \in E} I_e$ will be a solution to the problem posed in the lemma.
By assumption (2) and the proposition $\mathcal{C}/U_i$ has enough points. The points of $\mathcal{C}/U_i$ give points of $\mathcal{C}$ via the procedure of Lemma \ref{lemma-point-morphism-sites}. Thus it suffices to show: if $\phi : \mathcal{F} \to \mathcal{G}$ is a map of sheaves on $\mathcal{C}$ such that $\phi|_{\mathcal{C}/U_i}$ is an isomorphism for all $i$, then $\phi$ is an isomorphism. By assumption (1) for every object $W$ of $\mathcal{C}$ there is a covering $\{W_j \to W\}_{j \in J}$ such that for $j \in J$ there is an $i \in I$ and a morphism $f_j : W_j \to U_i$. Then the maps $\mathcal{F}(W_j) \to \mathcal{G}(W_j)$ are bijective and similarly for $\mathcal{F}(W_j \times_W W_{j'}) \to \mathcal{G}(W_j \times_W W_{j'})$. The sheaf condition tells us that $\mathcal{F}(W) \to \mathcal{G}(W)$ is bijective as desired.
Assume (1) and let $\mathcal{F} \to \mathcal{G}$ be a surjective map of sheaves of sets. For $s \in \mathcal{G}(U)$ there exists a covering $\{U_i \to U\}$ and $t_i \in \mathcal{F}(U_i)$ mapping to $s|_{U_i}$, see Definition \ref{definition-sheaves-injective-surjective}. Think of $t_i$ as a map $t_i : h_{U_i}^\# \to \mathcal{F}$ via (\ref{equation-map-representable-into-sheaf}). Then precomposing $\coprod t_i : \coprod h_{U_i}^\# \to \mathcal{F}$ with the map $h_U^\# \to \coprod h_{U_i}^\#$ we get from (1) we obtain a section $t \in \mathcal{F}(U)$ mapping to $s$. Thus (2) holds. \medskip\noindent Assume (2) holds. Let $\{U_i \to U\}$ be a covering. Then $\coprod h_{U_i}^\# \to h_U^\#$ is surjective (Lemma \ref{lemma-covering-surjective-after-sheafification}). Hence by (2) there exists a section $s$ of $\coprod h_{U_i}^\#$ mapping to the section $\text{id}_U$ of $h_U^\#$. This section corresponds to a map $h_U^\# \to \coprod h_{U_i}^\#$ which is right inverse to the sheafification of $\coprod h_{U_i} \to h_U$ which proves (1).
For any sheaf $\mathcal{F}$ we have $\mathcal{F}(U) = \Mor_{\Sh(\mathcal{C})}(h_U^\#, \mathcal{F})$. Hence, we see that (1) and (2) are equivalent. It is clear that (2) implies (3) implies (4). If every covering of $U$ is given by a nonempty family, then $\emptyset^+(U)$ is empty by definition of the plus construction. Note that $\emptyset^+ = \emptyset^\#$ as $\emptyset$ is a separated presheaf, see Theorem \ref{theorem-plus}. Thus we see that (4) implies (5). If (5) holds, then $\mathcal{F}(U)$ is a singleton for every sheaf $\mathcal{F}$ by the sheaf condition for $\mathcal{F}$, see Remark \ref{remark-sheaf-condition-empty-covering}. Thus (5) implies (2) and (1) -- (5) are equivalent. The final assertion of the lemma follows from Axiom (3) of Definition \ref{definition-site} applied the empty covering of $U$.
Let $U \in \Ob(\mathcal{C})$. We have to show that $(u^p\mathcal{G})^\#(U) = u^p(\mathcal{G}^\#)(U)$. It suffices to show that $(u^p\mathcal{G})^+(U) = u^p(\mathcal{G}^+)(U)$ since $\mathcal{G}^+$ is another presheaf for which the assumption of the lemma holds. We have $$ u^p(\mathcal{G}^+)(U) = \mathcal{G}^+(u(U)) = \colim_\mathcal{V} \check H^0(\mathcal{V}, \mathcal{G}) $$ where the colimit is over the coverings $\mathcal{V}$ of $u(U)$ in $\mathcal{D}$. On the other hand, we see that $$ u^p(\mathcal{G})^+(U) = \colim_\mathcal{U} \check H^0(u(\mathcal{U}), \mathcal{G}) $$ where the colimit is over the category of coverings $\mathcal{U} = \{U_i \to U\}_{i \in I}$ of $U$ in $\mathcal{C}$ and $u(\mathcal{U}) = \{u(U_i) \to u(U)\}_{i \in I}$. The condition that $u$ is continuous means that each $u(\mathcal{U})$ is a covering. Write $I = I_1 \amalg I_2$, where $$ I_2 = \{i \in I \mid u(U_i)\text{ is sheaf theoretically empty}\} $$ Then $u(\mathcal{U})' = \{u(U_i) \to u(U)\}_{i \in I_1}$ is still a covering of because each of the other pieces can be covered by the empty family and hence can be dropped by Axiom (2) of Definition \ref{definition-site}. Moreover, $\check H^0(u(\mathcal{U}), \mathcal{G}) = \check H^0(u(\mathcal{U})', \mathcal{G})$ by our assumption on $\mathcal{G}$. Finally, the condition that $u$ is almost cocontinuous implies that for every covering $\mathcal{V}$ of $u(U)$ there exists a covering $\mathcal{U}$ of $U$ such that $u(\mathcal{U})'$ refines $\mathcal{V}$. It follows that the two colimits displayed above have the same value as desired.
Let $\mathcal{I}$ be a finite connected index category. Let $\mathcal{I} \to \Sh(\mathcal{D})$, $i \mapsto \mathcal{G}_i$ by a diagram. We know that the colimit of this diagram is the sheafification of the colimit in the category of presheaves, see Lemma \ref{lemma-colimit-sheaves}. Denote $\colim^{Psh}$ the colimit in the category of presheaves. Since $\mathcal{I}$ is finite and connected we see that $\colim^{Psh}_i \mathcal{G}_i$ is a presheaf satisfying the assumptions of Lemma \ref{lemma-almost-cocontinuous-sheafification} (because a finite connected colimit of singleton sets is a singleton). Hence that lemma gives \begin{align*} u^s(\colim_i \mathcal{G}_i) & = u^s((\colim^{Psh}_i \mathcal{G}_i)^\#) \\ & = (u^p(\colim^{Psh}_i \mathcal{G}_i))^\# \\ & = (\colim^{PSh}_i u^p(\mathcal{G}_i))^\# \\ & = \colim_i u^s(\mathcal{G}_i) \end{align*} as desired.
We have seen in Lemma \ref{lemma-localize-topos} that $\Sh(\mathcal{C})/\mathcal{F}$ is a topos. In fact, we recall the proof. First we apply Lemma \ref{lemma-topos-good-site} to see that we may assume $\mathcal{C}$ is a site with a subcanonical topology, fibre products, a final object $X$, and an object $U$ with $\mathcal{F} = h_U$. The proof of Lemma \ref{lemma-localize-topos} shows that the morphism of topoi $j_\mathcal{F} : \Sh(\mathcal{C})/\mathcal{F} \to \Sh(\mathcal{C})$ is equal (modulo certain identifications) to the localization morphism $j_U : \Sh(\mathcal{C}/U) \to \Sh(\mathcal{C})$. \medskip\noindent Assume (2). This means that $j_U^{-1}j_{U, *}\mathcal{G} \to \mathcal{G}$ is an isomorphism for all sheaves $\mathcal{G}$ on $\mathcal{C}/U$. For any object $Z/U$ of $\mathcal{C}/U$ we have $$ (j_{U, *}h_{Z/U})(U) = \Mor_{\mathcal{C}/U}(U \times_X U/U, Z/U) $$ by Lemma \ref{lemma-localize-given-products}. Setting $\mathcal{G} = h_{Z/U}$ in the equality above we obtain $$ \Mor_{\mathcal{C}/U}(U \times_X U/U, Z/U) = \Mor_{\mathcal{C}/U}(U, Z/U) $$ for all $Z/U$. By Yoneda's lemma (Categories, Lemma \ref{categories-lemma-yoneda}) this implies $U \times_X U = U$. By Categories, Lemma \ref{categories-lemma-characterize-mono-epi} $U \to X$ is a monomorphism, in other words (1) holds. \medskip\noindent Assume (1). Then $j_U^{-1} j_{U, *} = \text{id}$ by Lemma \ref{lemma-restrict-back}.
We apply Lemma \ref{lemma-topos-good-site} to see that we may assume $\mathcal{C}$ is a site with the properties listed in that lemma. In particular $\mathcal{C}$ has a final object $X$ (so that $* = h_X$) and an object $U$ with $\mathcal{F} = h_U$. \medskip\noindent Let $\mathcal{D} = \mathcal{C}$ as a category but a covering is a family $\{V_j \to V\}$ of morphisms such that $\{V_i \to V\} \cup \{U \times_X V \to V\}$ is a covering. By our choice of $\mathcal{C}$ this means exactly that $$ h_{U \times_X V} \amalg \coprod h_{V_i} \longrightarrow h_V $$ is surjective. We claim that $\mathcal{D}$ is a site, i.e., the coverings satisfy the conditions (1), (2), (3) of Definition \ref{definition-site}. Condition (1) holds. For condition (2) suppose that $\{V_i \to V\}$ and $\{V_{ij} \to V_i\}$ are coverings of $\mathcal{D}$. Then the composition $$ \coprod \left( h_{U \times_X V_i} \amalg \coprod h_{V_{ij}} \right) \longrightarrow h_{U \times_X V} \amalg \coprod h_{V_i} \longrightarrow h_V $$ is surjective. Since each of the morphisms $U \times_X V_i \to V$ factors through $U \times_X V$ we see that $$ h_{U \times_X V} \amalg \coprod h_{V_{ij}} \longrightarrow h_V $$ is surjective, i.e., $\{V_{ij} \to V\}$ is a covering of $V$ in $\mathcal{D}$. Condition (3) follows similarly as a base change of a surjective map of sheaves is surjective. \medskip\noindent Note that the (identity) functor $u : \mathcal{C} \to \mathcal{D}$ is continuous and commutes with fibre products and final objects. Hence we obtain a morphism $f : \mathcal{D} \to \mathcal{C}$ of sites (Proposition \ref{proposition-get-morphism}). Observe that $f_*$ is the identity functor on underlying presheaves, hence fully faithful. To finish the proof we have to show that the essential image of $f_*$ is the full subcategory $E \subset \Sh(\mathcal{C})$ singled out in the lemma. To do this, note that $\mathcal{G} \in \Ob(\Sh(\mathcal{C}))$ is in $E$ if and only if $\mathcal{G}(U \times_X V)$ is a singleton for all objects $V$ of $\mathcal{C}$. Thus such a sheaf satisfies the sheaf property for all coverings of $\mathcal{D}$ (argument omitted). Conversely, if $\mathcal{G}$ satisfies the sheaf property for all coverings of $\mathcal{D}$, then $\mathcal{G}(U \times_X V)$ is a singleton, as in $\mathcal{D}$ the object $U \times_X V$ is covered by the empty covering.