Di you get the notes about ringed spaces?

I appreciate your answer =)

When you try to integrate some objects which appear in the context of Lie algebroids, VB-algebroids, etc. For example, in this pre-print arxiv.org/pdf/1608.00664.pdf the authors integrate VB-algebroids using strict 2-functors. For me, it appears when I try to integrate non-linear versions of VB-algebroids.

The problem boils down to establishing the bijections$\mathsf{Sh}(1, p+q)\times \mathsf{Sh}(p, q)\simeq \mathsf{Sh}(p+1, q)\times \mathsf{Sh}(1, p)$ and $\mathsf{Sh}(1, p+q)\times \mathsf{Sh}(p, q)\simeq \mathsf{Sh}(p, q+1)\times \mathsf{Sh}(1, q)$ and this really.

@DeaneYang maybe, I'll give it a try, thanks for the hint =)

The fiber on $(p, t)$ is $(A_M)_p\times \{t\}$ where $(A_M)_p$ is the fiber of $A_M$ over $p$. It is isomorphic to the pullback of $A_M$ by the projection on the first component $M\times I\longrightarrow M$

I've added some thoughts, could you check it?