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Lemma 7.19.2. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be cocontinuous. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Then ${}_pu\mathcal{F}$ is a sheaf on $\mathcal{D}$, which we will denote ${}_su\mathcal{F}$.

Proof. Let $\{V_j \to V\}_{j \in J}$ be a covering of the site $\mathcal{D}$. We have to show that $$ \xymatrix{ {}_pu\mathcal{F}(V) \ar[r] & \prod {}_pu\mathcal{F}(V_j) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod {}_pu\mathcal{F}(V_j \times_V V_{j'}) } $$ is an equalizer diagram. Since ${}_pu$ is right adjoint to $u^p$ we have $$ {}_pu\mathcal{F}(V) = \mathop{\rm Mor}\nolimits_{\textit{PSh}(\mathcal{D})}(h_V, {}_pu\mathcal{F}) = \mathop{\rm Mor}\nolimits_{\textit{PSh}(\mathcal{C})}(u^ph_V, \mathcal{F}) = \mathop{\rm Mor}\nolimits_{\textit{Sh}(\mathcal{C})}((u^ph_V)^\#, \mathcal{F}) $$ Hence it suffices to show that \begin{equation} \tag{7.19.2.1} \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 7.10.13.)

We first show that the second arrow of (7.19.2.1) becomes surjective after sheafification. To do this we use Lemma 7.12.2. 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.

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 (7.19.2.1). 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. $\square$

\begin{lemma}
\label{lemma-pu-sheaf}
Let $\mathcal{C}$ and $\mathcal{D}$ be sites.
Let $u : \mathcal{C} \to \mathcal{D}$ be cocontinuous.
Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$.
Then ${}_pu\mathcal{F}$ is a sheaf on $\mathcal{D}$,
which we will denote ${}_su\mathcal{F}$.
\end{lemma}

\begin{proof}
Let $\{V_j \to V\}_{j \in J}$ be a covering of the site $\mathcal{D}$.
We have to show that
$$
\xymatrix{
{}_pu\mathcal{F}(V) \ar[r] &
\prod {}_pu\mathcal{F}(V_j) \ar@<1ex>[r] \ar@<-1ex>[r] &
\prod {}_pu\mathcal{F}(V_j \times_V V_{j'})
}
$$
is an equalizer diagram. Since ${}_pu$ is right adjoint to $u^p$
we have
$$
{}_pu\mathcal{F}(V) =
\Mor_{\textit{PSh}(\mathcal{D})}(h_V, {}_pu\mathcal{F}) =
\Mor_{\textit{PSh}(\mathcal{C})}(u^ph_V, \mathcal{F}) =
\Mor_{\textit{Sh}(\mathcal{C})}((u^ph_V)^\#, \mathcal{F})
$$
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.
\end{proof}

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