I am trying to understand cocycle description of gerbes as in <https://arxiv.org/pdf/math/0611317.pdf>.


Let $\mathcal{P}$ be a gerbe on a topological space $X$ i.e., $\mathcal{P}$ is a stack over groupoids that is both locally non empty and locally connected. 


As $\mathcal{P}$ is locally nonempty, we can choose an open cover $\mathcal{U}=\{U\_i\}$ such that $\mathcal{P}(U\_i)\neq \emptyset$.


So, we can choose $x\_i\in \mathcal{P}(U\_i)$. As $\mathcal{P}$ is a stack, we get a sheaf over $U\_i$, $G\_i=\underline{\text{Aut}}(x\_i)$ over $U\_i$.


Given any sheaf on a space $X$, one can associate a group bundle (Etale space) over $X$. I guess after identifying sheaf with etale space, they are calling $G\_i$ to be bundle of groups on $U\_i$ (pg no $10$).


Once we chose $x\_i$ as above, we also choose arrows $\phi\_{ij}:x\_j|\_{U\_{ij}}\rightarrow x\_i|\_{U\_{ij}}$ in $\mathcal{P}(U\_{ij})$.


What I don’t understand is the map $\lambda\_{ij}:G\_j|\_{U\_{ij}}\rightarrow G\_i|\_{U\_{ij}}$. I fail to see how they are identifying elements of $G\_j|\_{U\_{ij}}$ as just maps $x\_j\rightarrow x\_j$. They are using notion of sheaf and that of etale space simultaneously and it is getting confused.


Any suggestion is welcome. 



Cocycle description of gerbes
It might help to divide this process into finer steps. We choose our open cover so that the groupoids $\mathcal{P}(U\_i)$ and $\mathcal{P}(U\_i \cap U\_j)$ are connected and nonempty. Choose objects $x\_i \in \mathcal{P}(U\_i)$ and $x\_{i,j} \in \mathcal{P}(U\_i \cap U\_j)$, and let $G\_i = \operatorname{Aut}(x\_i)$ and $G\_{i,j} = \operatorname{Aut}(x\_{i,j})$. By the fibered category property, the inclusions $\iota\_{i,j}: U\_i \cap U\_j \to U\_i$ yield restriction functors $\iota\_{i,j}^\*: \mathcal{P}(U\_i) \to \mathcal{P}(U\_i \cap U\_j)$. By the connected property, there exist isomorphisms $\iota\_{i,j}^\sharp: \iota\_{i,j}^\*(x\_i) \to x\_{i,j}$ in $\mathcal{P}(U\_i \cap U\_j)$, and any such choice yields a corresponding isomorphism of groups $\iota\_{i,j}^\flat: \iota\_{i,j}^\*(G\_i) \to G\_{i,j}$. If you choose your isomorphisms $\iota\_{i,j}^\sharp$, giving a choice of
$$\phi\_{ij} = (\iota\_{i,j}^\sharp)^{-1} \circ \iota\_{j,i}^\sharp$$
then you automatically get 
$$\lambda\_{ij} = (\iota\_{i,j}^\flat)^{-1} \circ \iota\_{j,i}^\flat$$.


