I am reading Differentiable stacks and gerbes by Kai Behrend and Ping Xu.
Let $\mathfrak{S}$ denote the category of smooth manifolds and smooth maps. Consider Grothendieck topology given by open covers $\{U_i\rightarrow X\}$.
The result I am trying to understand is the following.
Let $F$ be a sheaf over $\mathfrak{S}$. Let $X$ be a manifold and $F\rightarrow X$ be a morphism. Suppose that $\{U_i\rightarrow X\}$ is a cover for $X$ and that for every $i$, the sheaf $F_i=U_i\times_XF$ is representable. Then $F$ is representable.
A sheaf $F:\mathfrak{S}^{op}\rightarrow (Set)$ defines a groupoid fibration (category fibered in groupoids $\mathcal{D}$) over $\mathfrak{S}$ as follows. Objects of $\mathcal{D}$ are $(U,x)$ such that $U$ is an object of $\mathfrak{S}$ and $x\in F(U)$. Given objects $(U,x)$ and $(V,y)$, a morphism $(U,x)\rightarrow (V,y)$ consists of a smooth map $f:U\rightarrow V$ such that $F(f):F(V)\rightarrow F(U)$ takes $y$ to $x$ i.e., $F(f)(y)=x$.
As $F_i$ is representable, there exists a manifold $Y_i$ representing $F_i$. I am trying to use that $F$ is a sheaf to glue these manifolds $Y_i$ to produce a manifold $Y$ and an isomorphism $F\rightarrow Y$ without success.
Any hints are welcome.
