I am looking for some examples of gerbes over stacks (as defined in [Understanding definition of gerbe over a stack](https://mathoverflow.net/questions/307123/understanding-definition-of-gerbe-over-a-stack)) that comes from manifolds.


Let $M$ be a manifold then $\underline{M}$ is a stack associated to $M$ (I can give details if any one needs).


We know when to call a map of stacks $\mathcal{D}\rightarrow \mathcal{C}$ a gerbe over stack. Suppose $\mathcal{D}$ is coming from a manifold $M$ and $\mathcal{C}$ is coming from a manifold $N$ i.e., we have $\underline{M}\rightarrow \underline{N}$.


Given a map of stacks $F:\underline{M}\rightarrow \underline{N}$, we have unique map $F(id):M\rightarrow N$ associated to $F$.


What can we say about this map? Any comments are welcome.


First condition says that $F:\underline{M}\rightarrow \underline{N}$ is an epimorphism. Suppose $f:M\rightarrow N$ is a surjective submersion, then it follows that $F:\underline{M}\rightarrow \underline{N}$ is an epimorphism.


Second condition says that the diagonal map $\underline{M}\rightarrow \underline{M}\times\_{\underline{N}}\underline{M}$ is an epimorphism. Suppose diagonal map $M\rightarrow M\times\_N M$ is a surjective submersion, then $\underline{M}\rightarrow \underline{M}\times\_{\underline{N}}\underline{M}$ is an epimorphism.


But then, $M\rightarrow M\times\_N M$ is surjective implies that $f$ is injective. 


An injective, surjectve submersion is a diffeomorphism. So, I landed up with a diffeomorphism conidtion on $M\rightarrow N$.


Do we have other non trivial examples?


Edit: Ycor has added algebraic geometry tag. I am also ok with algebraic geometry examples. I do not know much (not even the precise definition in that set up). Smooth manifolds are replaced by schemes (I guess). Given a scheme, there is an associated stack (algebraic stack??). Gerbe over stack in case of algebraic geometry of the form $\underline{X}\rightarrow \underline{Y}$ should correspond to some morphism of schemes $X\rightarrow Y$. It will be good if you can define gerbe over stack in algebraic geometry set up (loosely atleast) and the give examples.



Examples of of gerbe over stacks in terms of manifolds
There are no other such gerbes. If $M$ and $N$ are manifolds, and $p\colon \underline{M}\to \underline{N}$ is a gerbe, then the corresponding map of manifolds is a diffeomorphism. The same holds if one merely has representable stacks, rather than something of the form $\underline{M}$ etc.


