A morphism of stacks \(F:\mathcal{D}\rightarrow \mathcal{C}\) is said to be a gerbe over stack if following two conditions hold :
  1. Given a manifold \(U\) and an object \(\xi\in \mathcal{C}(U)\), there exists a covering \(\{U_i\rightarrow U\}\) (depending on the Grothendieck topology that we have fixed on the category \(Man\) of manifolds) and objects \(x_i\in \mathcal{D}(U_i)\) with an isomorphism \(F(x_i)\rightarrow \xi|_{U_i}\) for each \(i\).
  2. Given a manifold \(U\) and an arrow \(\xi\rightarrow \eta\) in \(\mathcal{C}(U)\), there exists a covering \(\{U_i\rightarrow U\}\) (depending on the Grothendieck topology that we have fixed on the category \(Man\) of manifolds) and arrows \(x_i\rightarrow y_i\) in \(\mathcal{D}(U_i)\) such that