Criterion for a sheaf $\mathfrak{S}^{op}\rightarrow (Set)$ to be representable

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.

• Did you consider the fact that multiple intersections are also representable (since open embeddings are representable maps) to construct the gluing? – Denis Nardin Jan 2 at 8:53
• By multiple intersections you mean $U_{ij}\times_X F$?? @DenisNardin – Praphulla Koushik Jan 2 at 9:20
• Indeed, precisely. – Denis Nardin Jan 2 at 10:01
• @DenisNardin Yes. Even Kai Behrend paper tries in that way but I am not very happy in proceeding that way.. So, trying to come up with something on my own (with help from this site as I already posted the question :D) – Praphulla Koushik Jan 2 at 10:16