Let $\mathcal{C}$ be a site and $f:\mathcal{F}\to \mathcal{G}$ a morphism of $2$-sheaves. According to <https://mathoverflow.net/q/307366>, this is an epimorphism if and only if it is *almost surjective*, that is to say if for all $U$, and any $g\in G(U)$, there exists an open cover $\{U\_i\to U\}$ such that there exists $x\_i\in F(U\_i)$ such that $f(x\_i)\cong g\rvert\_{U\_i}$. If we assume that our $2$-topos has enough points, can we check this on stalks, that is to say on the pullback of $f$ to all points of our $2$-topos?



Do stalks see epimorphism of stacks?
By definition (e.g. Remark 6.5.4.7 of *Higher topos theory*), an $n$-topos $\mathcal{E}$ has enough points if for every morphism $f:X\to Y$ in $\mathcal{E}$, whenever $p^\*(f)$ is an equivalence for all points $p:\mathcal{S}\_n \to \mathcal{E}$ (where $\mathcal{S}\_n$ is the $n$-topos of $(n-1)$-groupoids), then $f$ is already an equivalence.


What you call an "epimorphism" or "almost surjective" is perhaps better called an [effective epimorphism](https://ncatlab.org/nlab/show/effective+epimorphisms+in+an+%28%E2%88%9E%2C1%29-category), since it is a faithful generalization of that concept from 1-categories, whereas it is not really a faithful generalization of the concept of epimorphism (e.g. it is not the same as a monomorphism in the opposite $n$-category). By Prop. 7.2.1.14 of *Higher topos theory*, a morphism in an $n$-topos is an effective epimorphism precisely when its truncation to a morphism of 1-sheaves is an (effective) epimorphism in the corresponding 1-topos; this shows that the effective epimorphisms agree with your concrete description.


Now the effective epimorphisms in an $n$-topos are the left class of a [factorization system](https://ncatlab.org/nlab/show/%28n-connected%2C+n-truncated%29+factorization+system) whose right class are the monomorphisms. Thus, a morphism is an effective epimorphism precisely if the monomorphism half of this factorization is an equivalence. It follows that being an effective epimorphism is reflected by any conservative family of functors, such as the family of points of an $n$-topos with enough points.


