diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/sites.tex stacks-0.2/src/sites.tex
--- stacks-0.2.orig/src/sites.tex	Mon Oct 17 10:59:55 2005
+++ stacks-0.2/src/sites.tex	Mon Oct 31 07:38:06 2005
@@ -170,7 +170,7 @@
 \begin{definition}
 \label{definition-presheaf}
 A presheaf $\mathcal{F}$ on a category $\mathcal{C}$ with values in a category
-$\mathcal{A}$ is a contravariant functor from $\mathcal{C}$ to $\mathcal{C}$,
+$\mathcal{A}$ is a contravariant functor from $\mathcal{C}$ to $\mathcal{A}$,
 i.e., $\mathcal{F} : \mathcal{C}^\circ \to \mathcal{A}$.
 \end{definition}
 
@@ -263,13 +263,36 @@
 
 \begin{lemma}
 \label{lemma-indepent-refinement}
-Any two morphisms $f,f': \mathcal{U} \to \mathcal{V}$ of coverings
+Any two morphisms $f,g: \mathcal{U} \to \mathcal{V}$ of coverings
 inducing the same morphism $U \to V$ induce the same
-map $\mathcal{F}(\mathcal{U}) \to \mathcal{F}(\mathcal{V})$.
+map $\mathcal{F}(\mathcal{V}) \to \mathcal{F}(\mathcal{U})$.
 \end{lemma}
 
 \begin{proof}
-FIXME.
+Let $\mathcal{U}=\{U_i \to U\}_{i\in I}$ and 
+$\mathcal{V}=\{V_j \to V\}_{j\in J}$.
+The morphism~$f$ consists of a map $U\to V$, a map $\alpha\colon I\to J$ and
+maps $f_i\colon U_i\to V_{\alpha(i)}$.
+Likewise, $g$~determines a map $\beta\colon I\to J$ and maps
+$g_i\colon U_i\to V_{\beta(i)}$.
+As $f$~and~$g$ induce the same map $U\to V$, the diagram
+$$\xymatrix{&V_{\alpha(i)}\ar[dr]\\
+  U_i\ar[ur]^{f_i}\ar[dr]_{g_i}&&V\\
+  &V_{\beta(i)}\ar[ur]}$$
+is commutative for every~$i\in I$. Hence $f$~and~$g$ factor through 
+the fibre product
+$$\xymatrix{&V_{\alpha(i)}\\
+  U_i\ar[r]^-\varphi\ar[ur]^{f_i}\ar[dr]_{g_i}&
+  V_{\alpha(i)}\times_VV_{\beta(i)}\ar[u]_{\text{pr}_1}\ar[d]^{\text{pr}_2}\\
+  &V_{\beta(i)}.}$$
+Now let $s=(s_j)_j\in\mathcal{F}(\mathcal{V})$.
+Then for all~$i\in I$:
+ $$(f^*s)_i=f_i^*(s_{\alpha(i)})=\varphi^*\text{pr}_1^*(s_{\alpha(i)})
+   =\varphi^*\text{pr}_2^*(s_{\beta(i)})=g_i^*(s_{\beta(i)})=(g^*s)_i,$$
+where the middle equality is given by the definition 
+of~$\mathcal{F}(\mathcal{V})$.
+This shows that the maps $\mathcal{F}(\mathcal{V})\to\mathcal{F}(\mathcal{U})$
+induced by $f$~and~$g$ are equal.
 \end{proof}
 
 \noindent