hcju/stacksqa huggingface dataset:
Parts (1), (2), (3), and (4) are straightforward consequences of the definitions and the fact that the functor $j$ commutes with fibre products. We omit the details. To see (5) recall that $f_*$ is given by ${}_su = {}_pu$. Hence the value of $j_V^{-1}f_*\mathcal{F}$ on $V'/V$ is the value of ${}_pu\mathcal{F}$ on $V'$ which is the limit of the values of $\mathcal{F}$ on the category ${}^u_{V'}\mathcal{I}$. Clearly, there is an equivalence of categories $$ {}^u_{V'}\mathcal{I} \to {}^{u'}_{V'/V}\mathcal{I} $$ Since the value of $f'_*j^{-1}\mathcal{F}$ on $V'/V$ is given by the limit of the values of $j^{-1}\mathcal{F}$ on the category ${}^{u'}_{V'/V}\mathcal{I}$ and since the values of $j^{-1}\mathcal{F}$ on objects of ${}^u_V\mathcal{I}$ are just the values of $\mathcal{F}$ (by Lemma \ref{lemma-when-shriek} as $j$ is continuous and cocontinuous) we see that (5) is true.
The categories ${}^{u'}_{V'}\mathcal{I}$ and ${}^{\ \ \ u}_{v(V')}\mathcal{I}$ are defined in Section \ref{section-more-functoriality-PSh}. The functor in condition (4) sends the object $\psi : u'(U') \to V'$ of ${}^{u'}_{V'}\mathcal{I}$ to the object $v(\psi) : uv'(U') = vu'(U') \to v(V')$ of ${}^{\ \ \ u}_{v(V')}\mathcal{I}$. Recall that $g^{-1}$ is given by $v^p$ (Lemma \ref{lemma-when-shriek}) and $f_*$ is given by ${}_su = {}_pu$. Hence the value of $g^{-1}f_*\mathcal{F}$ on $V'$ is the value of ${}_pu\mathcal{F}$ on $v(V')$ which is the limit $$ \lim_{u(U) \to v(V') \in \Ob({}^{\ \ \ u}_{v(V')}\mathcal{I}^{opp})} \mathcal{F}(U) $$ By the same reasoning, the value of $f'_*(g')^{-1}\mathcal{F}$ on $V'$ is given by the limit $$ \lim_{u'(U') \to V' \in \Ob({}^{u'}_{V'}\mathcal{I}^{opp})} \mathcal{F}(v'(U')) $$ Thus assumption (4) and Categories, Lemma \ref{categories-lemma-initial} show that these agree and the first equality of the lemma is proved. The second equality follows from the first by uniqueness of adjoints.
Namely, we have $$ f'_*(g')^{-1}\mathcal{F} = (u')^p((v')^p\mathcal{F})^\# = (u')^p(v')^p\mathcal{F} $$ The first equality by definition and the second by Lemma \ref{lemma-when-shriek}. We have $$ g^{-1}f_*\mathcal{F} = (v^pu^p\mathcal{F})^\# = ((u')^p(v')^p\mathcal{F})^\# = (u')^p(v')^p\mathcal{F} $$ The first equality by definition, the second because $u \circ v = v' \circ u'$, the third because we already saw that $(u')^p(v')^p\mathcal{F}$ is a sheaf. This proves $f'_* \circ (g')^{-1} = g^{-1} \circ f_*$ and the equality $g'_! \circ (f')^{-1} = f^{-1} \circ g_!$ follows by uniqueness of left adjoints.
We have seen the existence and commutativity of the diagrams in Lemma \ref{lemma-localize-cocontinuous}. We have to check hypotheses (1) -- (5) of Lemma \ref{lemma-equivalence} for the induced functor $u : \mathcal{C}/U \to \mathcal{D}/u(U)$. This is completely mechanical. \medskip\noindent Property (1). This is Lemma \ref{lemma-localize-cocontinuous}. \medskip\noindent Property (2). Let $\{U_i'/U \to U'/U\}_{i \in I}$ be a covering of $U'/U$ in $\mathcal{C}/U$. Because $u$ is continuous we see that $\{u(U_i')/u(U) \to u(U')/u(U)\}_{i \in I}$ is a covering of $u(U')/u(U)$ in $\mathcal{D}/u(U)$. Hence (2) holds for $u : \mathcal{C}/U \to \mathcal{D}/u(U)$. \medskip\noindent Property (3). Let $a, b : U''/U \to U'/U$ in $\mathcal{C}/U$ be morphisms such that $u(a) = u(b)$ in $\mathcal{D}/u(U)$. Because $u$ satisfies (3) we see there exists a covering $\{f_i : U''_i \to U''\}$ in $\mathcal{C}$ such that $a \circ f_i = b \circ f_i$. This gives a covering $\{f_i : U''_i/U \to U''/U\}$ in $\mathcal{C}/U$ such that $a \circ f_i = b \circ f_i$. Hence (3) holds for $u : \mathcal{C}/U \to \mathcal{D}/u(U)$. \medskip\noindent Property (4). Let $U''/U, U'/U \in \Ob(\mathcal{C}/U)$ and a morphism $c : u(U'')/u(U) \to u(U')/u(U)$ in $\mathcal{D}/u(U)$ be given. Because $u$ satisfies property (4) there exists a covering $\{f_i : U_i'' \to U''\}$ in $\mathcal{C}$ and morphisms $c_i : U_i'' \to U'$ such that $u(c_i) = c \circ u(f_i)$. We think of $U_i''$ as an object over $U$ via the composition $U_i'' \to U'' \to U$. It may not be true that $c_i$ is a morphism over $U$! But since $u(c_i)$ is a morphism over $u(U)$ we may apply property (3) for $u$ and find coverings $\{f_{ik} : U''_{ik} \to U''_i\}$ such that $c_{ik} = c_i \circ f_{ik} : U''_{ik} \to U'$ are morphisms over $U$. Hence $\{f_i \circ f_{ik} : U''_{ik}/U \to U''/U\}$ is a covering in $\mathcal{C}/U$ such that $u(c_{ik}) = c \circ u(f_{ik})$. Hence (4) holds for $u : \mathcal{C}/U \to \mathcal{D}/u(U)$. \medskip\noindent Property (5). Let $h : V \to u(U)$ be an object of $\mathcal{D}/u(U)$. Because $u$ satisfies property (5) there exists a covering $\{c_i : u(U_i) \to V\}$ in $\mathcal{D}$. By property (4) we can find coverings $\{f_{ij} : U_{ij} \to U_i\}$ and morphisms $c_{ij} : U_{ij} \to U$ such that $u(c_{ij}) = h \circ c_i \circ u(f_{ij})$. Hence $\{u(U_{ij})/u(U) \to V/u(U)\}$ is a covering in $\mathcal{D}/u(U)$ of the desired shape and we conclude that (5) holds for $u : \mathcal{C}/U \to \mathcal{D}/u(U)$.
Warning: Some of the statements above may look be a bit confusing at first; this is because objects of $\mathcal{C}'$ can also be viewed as sheaves on $\mathcal{C}$! We omit the proof that the coverings of $\mathcal{C}'$ as described in the lemma satisfy the conditions of Definition \ref{definition-site}. \medskip\noindent Suppose that $\{\mathcal{F}_i \to \mathcal{F}\}$ is a surjective family of morphisms of sheaves. Let $\mathcal{G}$ be another sheaf. Part (2) of the lemma says that the equalizer of $$ \xymatrix{ \Mor_{\Sh(\mathcal{C})}( \coprod_{i \in I} \mathcal{F}_i, \mathcal{G}) \ar@<1ex>[r] \ar@<-1ex>[r] & \Mor_{\Sh(\mathcal{C})}( \coprod_{(i_0, i_1) \in I \times I} \mathcal{F}_{i_0} \times_\mathcal{F} \mathcal{F}_{i_1}, \mathcal{G}) } $$ is $\Mor_{\Sh(\mathcal{C})}(\mathcal{F}, \mathcal{G}).$ This is clear (for example use Lemma \ref{lemma-coequalizer-surjection}). \medskip\noindent To prove (3) we have to check conditions (1) -- (5) of Lemma \ref{lemma-equivalence}. The fact that $v$ is cocontinuous is equivalent to the description of surjective maps of sheaves in Lemma \ref{lemma-mono-epi-sheaves}. The functor $v$ is continuous because $U \mapsto h_U^\#$ commutes with fibre products, and transforms coverings into coverings (see Lemma \ref{lemma-sheafification-exact}, and Lemma \ref{lemma-covering-surjective-after-sheafification}). Properties (3), (4) of Lemma \ref{lemma-equivalence} are statements about morphisms $f : h_{U'}^\# \to h_U^\#$. Such a morphism is the same thing as an element of $h_U^\#(U')$. Hence (3) and (4) are immediate from the construction of the sheafification. Property (5) of Lemma \ref{lemma-equivalence} is Lemma \ref{lemma-sheaf-coequalizer-representable}. Denote $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}')$ the equivalence of topoi associated with $v$ by Lemma \ref{lemma-equivalence}. \medskip\noindent Let $\mathcal{F}$ be as in part (4) of the lemma. For any $U \in \Ob(\mathcal{C})$ we have $$ g^{-1}h_\mathcal{F}(U) = h_\mathcal{F}(v(U)) = \Mor_{\Sh(\mathcal{C})}(h_U^\#, \mathcal{F}) = \mathcal{F}(U) $$ The first equality by Lemma \ref{lemma-when-shriek}. Thus part (4) holds. \medskip\noindent Let $\mathcal{F} \in \Ob(\mathcal{C}')$. Let $U \in \Ob(\mathcal{C})$. Then \begin{align*} g_*h_U^\#(\mathcal{F}) & = \Mor_{\Sh(\mathcal{C}')}(h_\mathcal{F}, g_*h_U^\#) \\ & = \Mor_{\Sh(\mathcal{C})}(g^{-1}h_\mathcal{F}, h_U^\#) \\ & = \Mor_{\Sh(\mathcal{C})}(\mathcal{F}, h_U^\#) \\ & = \Mor_{\mathcal{C}'}(\mathcal{F}, h_U^\#) \end{align*} as desired (where the third equality was shown above).
Consider the full subcategory $\mathcal{C}_1 \subset \Sh(\mathcal{C})$ consisting of all $h_U^\#$ for all $U \in \Ob(\mathcal{C})$, the given sheaves $\mathcal{F}_i$ and the final sheaf $*$ (see Example \ref{example-singleton-sheaf}). We are going to inductively define full subcategories $$ \mathcal{C}_1 \subset \mathcal{C}_2 \subset \mathcal{C}_2 \subset \ldots \subset \Sh(\mathcal{C}) $$ Namely, given $\mathcal{C}_n$ let $\mathcal{C}_{n + 1}$ be the full subcategory consisting of all fibre products and subsheaves of objects of $\mathcal{C}_n$. (Note that $\mathcal{C}_{n + 1}$ has a set of objects.) Set $\mathcal{C}' = \bigcup_{n \geq 1} \mathcal{C}_n$. A covering in $\mathcal{C}'$ is any family $\{\mathcal{G}_j \to \mathcal{G}\}_{j \in J}$ of morphisms of objects of $\mathcal{C}'$ such that $\coprod \mathcal{G}_j \to \mathcal{G}$ is surjective as a map of sheaves on $\mathcal{C}$. The functor $v : \mathcal{C} \to \mathcal{C'}$ is given by $U \mapsto h_U^\#$. Apply Lemma \ref{lemma-special-equivalence}.
Consider the full subcategory $\mathcal{C}_1 \subset \Sh(\mathcal{C})$ consisting of all $h_U^\#$ and all $f^{-1}h_V^\#$ for all $U \in \Ob(\mathcal{C})$ and all $V \in \Ob(\mathcal{D})$. Let $\mathcal{C}_{n + 1}$ be a full subcategory consisting of all fibre products of objects of $\mathcal{C}_n$. Set $\mathcal{C}' = \bigcup_{n \geq 1} \mathcal{C}_n$. A covering in $\mathcal{C}'$ is any family $\{\mathcal{F}_i \to \mathcal{F}\}_{i \in I}$ such that $\coprod_{i \in I} \mathcal{F}_i \to \mathcal{F}$ is surjective as a map of sheaves on $\mathcal{C}$. The functor $v : \mathcal{C} \to \mathcal{C'}$ is given by $U \mapsto h_U^\#$. The functor $u : \mathcal{D} \to \mathcal{C'}$ is given by $V \mapsto f^{-1}h_V^\#$. \medskip\noindent Part (1) follows from Lemma \ref{lemma-special-equivalence}. \medskip\noindent Proof of (2) and (3) of the lemma. The functor $u$ commutes with fibre products as both $V \mapsto h_V^\#$ and $f^{-1}$ do. Moreover, since $f^{-1}$ is exact and commutes with arbitrary colimits we see that it transforms a covering into a surjective family of morphisms of sheaves. Hence $u$ is continuous. To see that it defines a morphism of sites we still have to see that $u_s$ is exact. In order to do this we will show that $g^{-1} \circ u_s = f^{-1}$. Namely, then since $g^{-1}$ is an equivalence and $f^{-1}$ is exact we will conclude. Because $g^{-1}$ is adjoint to $g_*$, and $u_s$ is adjoint to $u^s$, and $f^{-1}$ is adjoint to $f_*$ it also suffices to prove that $u^s \circ g_* = f_*$. Let $U$ be an object of $\mathcal{C}$ and let $V$ be an object of $\mathcal{D}$. Then \begin{align*} (u^sg_*h_U^\#)(V) & = g_*h_U^\#(f^{-1}h_V^\#) \\ & = \Mor_{\Sh(\mathcal{C})}(f^{-1}h_V^\#, h_U^\#) \\ & = \Mor_{\Sh(\mathcal{D})}(h_V^\#, f_*h_U^\#) \\ & = f_*h_U^\#(V) \end{align*} The first equality because $u^s = u^p$. The second equality by Lemma \ref{lemma-special-equivalence} (5). The third equality by adjointness of $f_*$ and $f^{-1}$ and the final equality by properties of sheafification and the Yoneda lemma. We omit the verification that these identities are functorial in $U$ and $V$. Hence we see that we have $u^s \circ g_* = f_*$ for sheaves of the form $h_U^\#$. This implies that $u^s \circ g_* = f_*$ and we win (some details omitted).
Apply Lemma \ref{lemma-topos-good-site}. This means we may assume $\mathcal{C}$ is a site with subcanonical topology, and $\mathcal{F} = h_U = h_U^\#$ for some $U \in \Ob(\mathcal{C})$. Hence the material of Section \ref{section-localize} applies. In particular, there is an equivalence $\Sh(\mathcal{C}/U) = \Sh(\mathcal{C})/h_U^\#$ such that the composition $$ \Sh(\mathcal{C}/U) \to \Sh(\mathcal{C})/h_U^\# \to \Sh(\mathcal{C}) $$ is equal to $j_{U!}$, see Lemma \ref{lemma-essential-image-j-shriek}. Denote $a : \Sh(\mathcal{C})/h_U^\# \to \Sh(\mathcal{C}/U)$ the inverse functor, so $j_{\mathcal{F}!} = j_{U!} \circ a$, $j_\mathcal{F}^{-1} = a^{-1} \circ j_U^{-1}$, and $j_{\mathcal{F}, *} = j_{U, *} \circ a$. The description of $j_{\mathcal{F}!}$ follows from the above. The description of $j_\mathcal{F}^{-1}$ follows from Lemma \ref{lemma-compute-j-shriek-restrict}.
For any $\varphi : \mathcal{G} \to \mathcal{F}$, let us use the notation $\mathcal{G}/\mathcal{F}$ to denote the corresponding object of $\Sh(\mathcal{C})/\mathcal{F}$. We have $$ (j_{\mathcal{F}, *}(\mathcal{G}/\mathcal{F}))(U) = \Mor_{\Sh(\mathcal{C})}(h_U^\#, j_{\mathcal{F}, *}(\mathcal{G}/\mathcal{F})) = \Mor_{\Sh(\mathcal{C})/\mathcal{F}}(j_{\mathcal{F}}^{-1}h_U^{\#}, (\mathcal{G}/\mathcal{F})). $$ By Lemma \ref{lemma-localize-topos} this set is the fiber over the element $h_U^\# \times \mathcal{F} \to \mathcal{F}$ under the map of sets $$ \Mor_{\Sh(\mathcal{C})}(h_U^\# \times \mathcal{F}, \mathcal{G}) \xrightarrow{\varphi \circ} \Mor_{\Sh(\mathcal{C})}(h_U^\# \times \mathcal{F}, \mathcal{F}). $$ By the adjunction in Lemma \ref{lemma-internal-hom-sheaf}, we have \begin{align*} \Mor_{\Sh(\mathcal{C})}(h_U^{\#}\times\mathcal{F}, \mathcal{G}) & = \Mor_{\Sh(\mathcal{C})}(h_U^{\#},\SheafHom(\mathcal{F}, \mathcal{G})) \\ & = \Mor_{\Sh(\mathcal{C}/U)}(\mathcal{F}|_{\mathcal{C}/U}, \mathcal{G}|_{\mathcal{C}/U}), \\ \Mor_{\Sh(\mathcal{C})}(h_U^{\#} \times \mathcal{F}, \mathcal{F}) & = \Mor_{\Sh(\mathcal{C})}(h_U^{\#},\SheafHom(\mathcal{F},\mathcal{F})) \\ & = \Mor_{\Sh(\mathcal{C}/U)}(\mathcal{F}|_{\mathcal{C}/U}, \mathcal{F}|_{\mathcal{C}/U}), \end{align*} and under the adjunction, the map $\varphi\circ$ becomes the map $$ \Mor_{\Sh(\mathcal{C}/U)}(\mathcal{F}|_{\mathcal{C}/U}, \mathcal{G}|_{\mathcal{C}/U}) \longrightarrow \Mor_{\Sh(\mathcal{C}/U)}(\mathcal{F}|_{\mathcal{C}/U}, \mathcal{F}|_{\mathcal{C}/U}),\quad \psi \longmapsto \varphi|_{\mathcal{C}/U} \circ \psi, $$ the element $h_U^\# \times \mathcal{F} \to \mathcal{F}$ becomes $\text{id}_{\mathcal{F}|_{\mathcal{C}/U}}$. Therefore $(j_{\mathcal{F}, *}\mathcal{G}/\mathcal{F})(U)$ is isomorphic to the fiber of $\text{id}_{\mathcal{F}|_{\mathcal{C}/U}}$ under the map $$ \Mor_{\Sh(\mathcal{C}/U)}(\mathcal{F}|_{\mathcal{C}/U}, \mathcal{G}|_{\mathcal{C}/U}) \xrightarrow{\varphi|_{\mathcal{C}/U}\circ} \Mor_{\Sh(\mathcal{C}/U)}(\mathcal{F}|_{\mathcal{C}/U}, \mathcal{F}|_{\mathcal{C}/U}), $$ which is $\{\alpha : \mathcal{F}|_U \to \mathcal{G}|_U \text{ such that } \alpha \text{ is a right inverse to }\varphi|_U \}$ as desired.
We omit the verification that $\mathcal{C}/\mathcal{F}$ is a site and that $j$ is continuous and cocontinuous. By Lemma \ref{lemma-when-shriek} there exists a morphism of topoi $j$ as indicated, with $j^{-1}\mathcal{G}(U, s) = \mathcal{G}(U)$, and there is a left adjoint $j_!$ to $j^{-1}$. A morphism $\varphi : * \to j^{-1}\mathcal{G}$ on $\mathcal{C}/\mathcal{F}$ is the same thing as a rule which assigns to every pair $(U, s)$ a section $\varphi(s) \in \mathcal{G}(U)$ compatible with restriction maps. Hence this is the same thing as a morphism $\varphi : \mathcal{F} \to \mathcal{G}$ over $\mathcal{C}$. We conclude that $j_!* = \mathcal{F}$. In particular, for every $\mathcal{H} \in \Sh(\mathcal{C}/\mathcal{F})$ there is a canonical map $$ j_!\mathcal{H} \to j_!* = \mathcal{F} $$ i.e., we obtain a functor $j'_! : \Sh(\mathcal{C}/\mathcal{F}) \to \Sh(\mathcal{C})/\mathcal{F}$. An inverse to this functor is the rule which assigns to an object $\varphi : \mathcal{G} \to \mathcal{F}$ of $\Sh(\mathcal{C})/\mathcal{F}$ the sheaf $$ a(\mathcal{G}/\mathcal{F}) : (U, s) \longmapsto \{t \in \mathcal{G}(U) \mid \varphi(t) = s\} $$ We omit the verification that $a(\mathcal{G}/\mathcal{F})$ is a sheaf and that $a$ is inverse to $j'_!$.
We have seen in Lemma \ref{lemma-essential-image-j-shriek} that the composition $\Sh(\mathcal{C}/U) \to \Sh(\mathcal{C})/h_U^\# \to \Sh(\mathcal{C})$ is $j_{U!}$. The functor $\Sh(\mathcal{C})/h_U^\# \to \Sh(\mathcal{C})$ is $j_{\mathcal{F}!}$ by Lemma \ref{lemma-localize-topos}. Hence $j_{\mathcal{F}!} = j_{U!}$ via the identification. So $j_\mathcal{F}^{-1} = j_U^{-1}$ (by adjointness) and so $j_{\mathcal{F}, *} = j_{U, *}$ (by adjointness again).
The description of $j^{-1}$ and $j_!$ comes from the description of those functors in Lemma \ref{lemma-localize-topos}. The equality of functors $j_{\mathcal{G}!} = j_{\mathcal{F}!} \circ j_!$ is clear from the description of these functors (as forgetful functors). By adjointness we also obtain the equalities $j_\mathcal{G}^{-1} = j^{-1} \circ j_\mathcal{F}^{-1}$, and $j_{\mathcal{G}, *} = j_{\mathcal{F}, *} \circ j_*$.
Since the statement is about topoi and does not refer to the underlying sites we may change sites at will. Hence by the discussion in Remark \ref{remark-morphism-topoi-comes-from-morphism-sites} we may assume that $f$ is given by a continuous functor $u : \mathcal{D} \to \mathcal{C}$ satisfying the assumptions of Proposition \ref{proposition-get-morphism} between sites having all finite limits and subcanonical topologies, and such that $\mathcal{G} = h_V$ for some object $V$ of $\mathcal{D}$. Then $\mathcal{F} = f^{-1}h_V = h_{u(V)}$ by Lemma \ref{lemma-pullback-representable-sheaf}. By Lemma \ref{lemma-localize-morphism} we obtain a commutative diagram of morphisms of topoi $$ \xymatrix{ \Sh(\mathcal{C}/U) \ar[r]_{j_U} \ar[d]_{f'} & \Sh(\mathcal{C}) \ar[d]^f \\ \Sh(\mathcal{D}/V) \ar[r]^{j_V} & \Sh(\mathcal{D}), } $$ and we have $f'_*j_U^{-1} = j_V^{-1}f_*$. By Lemma \ref{lemma-localize-compare} we may identify $j_\mathcal{F}$ and $j_U$ and $j_\mathcal{G}$ and $j_V$. The description of $(f')^{-1}$ is given in Lemma \ref{lemma-localize-morphism}.
This is not a complete triviality as the choice of morphism of sites giving rise to $f$ made in the proof of Lemma \ref{lemma-localize-morphism-topoi} may be different from the morphisms of sites given to us in the lemma. But in both cases the functor $(f')^{-1}$ is described by the same rule. Hence they agree and the associated morphism of topoi is the same. Some details omitted.
The commutativity of the first square follows from the commutativity of the diagram in Lemma \ref{lemma-relocalize-topos} and the commutativity of the diagram in Lemma \ref{lemma-localize-morphism-topoi}. The description of $f_s^{-1}$ follows on combining the descriptions of $(f')^{-1}$ in Lemma \ref{lemma-localize-morphism-topoi} with the description of $(j_{\mathcal{F}/f^{-1}\mathcal{G}})^{-1}$ in Lemma \ref{lemma-relocalize-topos}. The commutativity of the last square then follows from the equality $$ f^{-1}\mathcal{H} \times_{f^{-1}\mathcal{G}, s} \mathcal{F} \times_\mathcal{F} \mathcal{F}' = f^{-1}(\mathcal{H} \times_\mathcal{G} \mathcal{G}') \times_{f^{-1}\mathcal{G}', s'} \mathcal{F}' $$ which is formal.
The first assertion is immediate from the definition of a sheaf, assumptions (1) and (2), and the definition of $u^pE$. The second is a restatement of the adjointness of $u^p$ and the stalk functor (Lemma \ref{lemma-adjoint-point-push-stalk}) restricted to sheaves. The third assertion follows as, for any set $E$, we have $$ \text{Map}(\mathcal{F}_p, E) = \Mor_{\textit{PSh}(\mathcal{C})}(\mathcal{F}, u^pE) = \Mor_{\Sh(\mathcal{C})}(\mathcal{F}^\#, u^sE) = \text{Map}(\mathcal{F}^\#_p, E) $$ by the adjointness property of sheafification.
Proof of (1). By the above the functors $p_*$ and $p^{-1}$ are adjoint. The functor $p^{-1}$ is required to be exact by Definition \ref{definition-point}. Hence the conditions imposed in Definition \ref{definition-topos} are all satisfied and we see that (1) holds. \medskip\noindent Proof of (2). Let $\{U_i \to U\}$ be a covering of $\mathcal{C}$. Then $\coprod (h_{U_i})^\# \to h_U^\#$ is surjective, see Lemma \ref{lemma-covering-surjective-after-sheafification}. Since $p^{-1}$ is exact (by definition of a morphism of topoi) we conclude that $\coprod u(U_i) \to u(U)$ is surjective. This proves part (1) of Definition \ref{definition-point}. Sheafification is exact, see Lemma \ref{lemma-sheafification-exact}. Hence if $U \times_V W$ exists in $\mathcal{C}$, then $$ h_{U \times_V W}^\# =  h_U^\# \times_{h_V^\#} h_W^\# $$ and we see that $u(U \times_V W) = u(U) \times_{u(V)} u(W)$ since $p^{-1}$ is exact. This proves part (2) of Definition \ref{definition-point}. Let $p' = u$, and let $\mathcal{F}_{p'}$ be the stalk functor defined by Equation (\ref{equation-stalk}) using $u$. There is a canonical comparison map $c : \mathcal{F}_{p'} \to \mathcal{F}_p = p^{-1}\mathcal{F}$. Namely, given a triple $(U, x, \sigma)$ representing an element $\xi$ of $\mathcal{F}_{p'}$ we think of $\sigma$ as a map $\sigma : h_U^\# \to \mathcal{F}$ and we can set $c(\xi) = p^{-1}(\sigma)(x)$ since $x \in u(U) = p^{-1}(h_U^\#)$. By Lemma \ref{lemma-points-recover} we see that $(h_U)_{p'} = u(U)$. Since conditions (1) and (2) of Definition \ref{definition-point} hold for $p'$ we also have $(h_U^\#)_{p'} = (h_U)_{p'}$ by Lemma \ref{lemma-point-pushforward-sheaf}. Hence we have $$ (h_U^\#)_{p'} = (h_U)_{p'} = u(U) = p^{-1}(h_U^\#) $$ We claim this bijection equals the comparison map $c : (h_U^\#)_{p'} \to p^{-1}(h_U^\#)$ (verification omitted). Any sheaf on $\mathcal{C}$ is a coequalizer of maps of coproducts of sheaves of the form $h_U^\#$, see Lemma \ref{lemma-sheaf-coequalizer-representable}. The stalk functor $\mathcal{F} \mapsto \mathcal{F}_{p'}$ and the functor $p^{-1}$ commute with arbitrary colimits (as they are both left adjoints). We conclude $c$ is an isomorphism for every sheaf $\mathcal{F}$. Thus the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{p'}$ is isomorphic to $p^{-1}$ and we in particular see that it is exact. This proves condition (3) of Definition \ref{definition-point} holds and $p'$ is a point. The final assertion has already been shown above, since we saw that $p^{-1} = (p')^{-1}$.
Part (1) we saw in Remark \ref{remark-pt-topos}. Moreover, recall that the equivalence associates to the set $E$ the sheaf $i_*E$ on $\mathcal{S}$ defined by the rule $V \mapsto \Mor_{\textit{Sets}}(V, E)$. Part (2) is clear from the definition of a point of $\mathcal{C}$ (Definition \ref{definition-point}) and the definition of a morphism of sites (Definition \ref{definition-morphism-sites}). Finally, consider $f_*i_*E$. By construction we have $$ f_*i_*E(U) = i_*E(u(U)) = \Mor_{\textit{Sets}}(u(U), E) $$ which is equal to $p_*E(U)$, see Equation (\ref{equation-skyscraper}). This proves (3).
Because $p_*$ is a right adjoint it commutes with arbitrary limits and it is left exact. The fact that $p^{-1}p_*E \to E$ is a canonically split surjection implies that $p_*$ is faithful, reflects injections, reflects surjections, and reflects isomorphisms. By Lemma \ref{lemma-point-site-topos} we may assume that $p$ comes from a point $u : \mathcal{C} \to \textit{Sets}$ of the underlying site $\mathcal{C}$. In this case the sheaf $p_*E$ is given by $$ p_*E(U) = \Mor_{\textit{Sets}}(u(U), E) $$ see Equation (\ref{equation-skyscraper}) and Definition \ref{definition-pushforward-point}. It follows immediately from this formula that $p_*$ transforms surjections into surjections and coequalizers into coequalizers.
Let $\mathcal{I} \to \Sh(\mathcal{C})$, $i \mapsto \mathcal{F}_i$ be a finite diagram of sheaves. We have to show that the stalk of the limit of this system agrees with the limit of the stalks. Let $\mathcal{F}$ be the limit of the system as a {\it presheaf}. According to Lemma \ref{lemma-limit-sheaf} this is a sheaf and it is the limit in the category of sheaves. Hence we have to show that $\mathcal{F}_p = \lim_\mathcal{I} \mathcal{F}_{i, p}$. Recall also that $\mathcal{F}$ has a simple description, see Section \ref{section-limits-colimits-PSh}. Thus we have to show that $$ \lim_i \colim_{\{(U, x)\}^{opp}} \mathcal{F}_i(U) = \colim_{\{(U, x)\}^{opp}} \lim_i \mathcal{F}_i(U). $$ This holds, by Categories, Lemma \ref{categories-lemma-directed-commutes}, because the opposite of the category of neighbourhoods is filtered by assumption.