Lemma 7.41.2. Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites associated to the continuous functor $u : \mathcal{C} \to \mathcal{D}$. Assume that for any object $U$ of $\mathcal{C}$ and any covering $\{ V_ j \to u(U)\} $ in $\mathcal{D}$ there exists a covering $\{ U_ i \to U\} $ in $\mathcal{C}$ such that the map of sheaves
\[ \coprod h_{u(U_ i)}^\# \to h_{u(U)}^\# \]
factors through the map of sheaves
\[ \coprod h_{V_ j}^\# \to h_{u(U)}^\# . \]
Then $f_*$ transforms surjective maps of sheaves into surjective maps of sheaves.
Proof.
Let $a : \mathcal{F} \to \mathcal{G}$ be a surjective map of sheaves on $\mathcal{D}$. Let $U$ be an object of $\mathcal{C}$ and let $s \in f_*\mathcal{G}(U) = \mathcal{G}(u(U))$. By assumption there exists a covering $\{ V_ j \to u(U)\} $ and sections $s_ j \in \mathcal{F}(V_ j)$ with $a(s_ j) = s|_{V_ j}$. Now we may think of the sections $s$, $s_ j$ and $a$ as giving a commutative diagram of maps of sheaves
\[ \xymatrix{ \coprod h_{V_ j}^\# \ar[r]_-{\coprod s_ j} \ar[d] & \mathcal{F} \ar[d]^ a \\ h_{u(U)}^\# \ar[r]^ s & \mathcal{G} } \]
By assumption there exists a covering $\{ U_ i \to U\} $ such that we can enlarge the commutative diagram above as follows
\[ \xymatrix{ & \coprod h_{V_ j}^\# \ar[r]_-{\coprod s_ j} \ar[d] & \mathcal{F} \ar[d]^ a \\ \coprod h_{u(U_ i)}^\# \ar[r] \ar[ur] & h_{u(U)}^\# \ar[r]^ s & \mathcal{G} } \]
Because $\mathcal{F}$ is a sheaf the map from the left lower corner to the right upper corner corresponds to a family of sections $s_ i \in \mathcal{F}(u(U_ i))$, i.e., sections $s_ i \in f_*\mathcal{F}(U_ i)$. The commutativity of the diagram implies that $a(s_ i)$ is equal to the restriction of $s$ to $U_ i$. In other words we have shown that $f_*a$ is a surjective map of sheaves.
$\square$
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