I am reading Brylinski's Loop Spaces, Characteristic Classes, and Geometric Quantization.

## The Statement

Let $C$ be a gerbe on a space $X$ with "abelian" band $H$, $f: Y \to X$ a local homeomorphism (this context uses local homeomorphisms in place of open subset sets and covers of open sets). Given an object $P$ of $C(Y)$, and an $H$ torsor $I$ over $Y$, there exists an object $Q$ of $C(Y)$ such that $I \overset{\sim}{\to} \underline{\text{Isom}}(P,Q)$ as $H$-torsors. The object $Q$ is called "the object obtained from $P$ by twisting by the H-torsor $I$".

## My Question

I can get a local ismorphism $I \overset{\sim}{\to} \underline{\text{Isom}}(P,Q)$ but how can I see it is, in fact, global?

## Some Details / A sketch of the proof

So you take a surjective local homeomorphism $g: Z \to Y$ so that there is a global section of $g^{-1}I$ (i.e. take a fine-enough open cover so that you have sections of $I$). Then pulling back this section $s$ under the maps $p_1, p_2: Z \times_Y Z \to Z$ we get two different sections of a sheaf which is a torsor under the pullback of $H$. So you get this section of the pullback of $H$ and since the pullback of $P$ is also a torsor under this sheaf, you have an automorphism of the pullback of $P$. This automorphism could also be thought of as an isomorphism $u: p_1^{-1}g^{-1}P \overset{\sim}{\to} p_2^{-1}g^{-1}P$ which satisfies the cocycle condition. Then by the descent property which this gerbe satisfies, we have an object $Q$ of $C(Y)$ with $\psi: g^{-1}Q \overset{\sim}{\to} g^{-1}P$.

So we started with a section of the pullback of $I$ (i.e a local section of $I$) and ended with an isomorphism between $P$ and $Q$'s pullbacks (i.e. a local isomorphism).

**What information can I use to show that this local isomorphism is in fact an isomorphism? Or is Brylinski simply abusing language?**

EDIT (2/22/13): I'm fairly certain I just need to show that on double intersections these local isomorphisms differ by an inner automorphism, but since my band is abelian this would say they glue together. I just haven't wrapped my head around this idea in the context of local homeomorphisms ans pullback diagrams quite yet.