Fix and add a proof

Thanks to Kestutis Cesnavicius
http://stacks.math.columbia.edu/tag/04WQ#comment-472

stacks.tex

@@ -735,14 +735,43 @@ \section{Stacks}
 F :
 \mathit{Mor}_{\mathcal{S}_1}(x, y)
 \longrightarrow
-\mathit{Mor}_{\mathcal{S}_2}(x, y)
+\mathit{Mor}_{\mathcal{S}_2}(F(x), F(y))
 $$
 is an isomorphism of sheaves on $\mathcal{C}/U$.
 \end{enumerate}
 \end{lemma}
 
 \begin{proof}
-Omitted.
+Assume (1). For $U, x, y$ as in (2) the displayed map $F$ evaluates to the map
+$F : \Mor_{\mathcal{S}_{1, V}}(x|_V, y|_V) \to
+\Mor_{\mathcal{S}_{2, V}}(F(x|_V), F(y|_V))$
+on an object $V$ of $\mathcal{C}$ lying over $U$.
+Now, since $F$ is fully faithful, the corresponding map
+$\Mor_{\mathcal{S}_1}(x|_V, y|_V) \to \Mor_{\mathcal{S}_2}(F(x|_V), F(y|_V))$
+is a bijection. Morphisms in the fibre category $\mathcal{S}_{1, V}$ are
+exactly those morphisms between $x|_V$ and $y|_V$ in $\mathcal{S}_1$ lying
+over $\text{id}_V$. Similarly, morphisms in the fibre category
+$\mathcal{S}_{2, V}$ are exactly those morphisms between $F(x|_V)$ and
+$F(y|_V)$ in $\mathcal{S}_2$ lying over $\text{id}_V$. Thus we find that $F$
+induces a bijection between these also. Hence (2) holds.
+
+\medskip\noindent
+Assume (2). Suppose given objects $U$, $V$ of $\mathcal{C}$ and
+$x \in \Ob(\mathcal{S}_{1, U})$ and
+$y \in \Ob(\mathcal{S}_{1, V})$. To show that $F$ is fully faithful,
+it suffices to prove it induces a bijection on
+morphisms lying over a fixed $f : U \to V$. Choose a strongly Cartesian
+$f^*y \to y$ in $\mathcal{S}_1$ lying above $f$. This results in a
+bijection between the set of morphisms $x \to y$ in $\mathcal{S}_1$ lying
+over $f$ and $\Mor_{\mathcal{S}_{1, U}}(x, f^*y)$. Since $F$ preserves
+strongly Cartesian morphisms as a $1$-morphism in the $2$-category
+of stacks over $\mathcal{C}$, we also get a bijection
+between the set of morphisms $F(x) \to F(y)$ in $\mathcal{S}_2$ lying
+over $f$ and $\Mor_{\mathcal{S}_{2, U}}(F(x), F(f^*y))$.
+Since $F$ induces a bijection
+$\Mor_{\mathcal{S}_{1, U}}(x, f^*y) \to
+\Mor_{\mathcal{S}_{2, U}}(F(x), F(f^*y))$
+we conclude (1) holds.
 \end{proof}
 
 \begin{lemma}