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 \emph{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 $f(x_i)\cong g\vert_{U_i}$. 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?

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?

Let $\mathcal{C}$ be a site and $f:\mathcal{F}\to \mathcal{G}$ a morphism of $n$-sheaves. If we assume that $\mathcal{C}$ is nice, then there exists a conservative family of fibre functors $\{\phi_i\}_{i\in I}$ such that $f$ is an equivalence if and only if for all $i\in I$, the induced morphism $\phi_i(f)$ is an equivalence. Does such a family also detect essential surjectivity, that is to say:

If for all $i\in I$, the $\phi_i(f)$ are essentially surjective, does it follow that for any $U\in \text{Ob}(\mathcal{C})$ the induced morphism $f(U):\text{Im}(f)(U)\to \mathcal{G}(U)$ is essentially surjective, where $\text{Im}(f)$ denotes the image $n$-sheaf? (I'm most interested in the $n=2$ case if that makes a difference).

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 \emph{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 $f(x_i)\cong g\vert_{U_i}$. 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?

Let $\mathcal{C}$ be a site and $f:\mathcal{F}\to \mathcal{G}$ a morphism of $n$-sheaves. If we assume that $\mathcal{C}$ is nice, then there exists a conservative family of fibre functors $\{\phi_i\}_{i\in I}$ such that $f$ is an isomorphism if and only if for all $i\in I$, the induced morphism $\phi_i(f)$ is an isomorphism. However, I feel I've always been told that isomorphisms of categories are the "wrong" thing to look out for, and that I'd rather should consider equivalence of categories. Hence I wonder if a conservative family of fibre functor can tell us something about this, that is to say:

If for all $i\in I$ the $\phi_i(f)$ are natural equivalences, does it follow that for any $U\in \text{Ob}(\mathcal{C})$ the induced morphism $f(U):\mathcal{F}(U)\to \mathcal{G}(U)$ is a natural equivalence?

Also, does a conservative family of fibre functors detect essential surjectivity, that is to say if for all $i\in I$, the $\phi_i(f)$ are essentially surjective, does it follow that for any $U\in \text{Ob}(\mathcal{C})$ the induced morphism $f(U):\text{Im}(f)(U)\to \mathcal{G}(U)$ is essentially surjective, where $\text{Im}(f)$ denotes the image $2$-sheaf?

Let $\mathcal{C}$ be a site and $f:\mathcal{F}\to \mathcal{G}$ a morphism of $n$-sheaves. If we assume that $\mathcal{C}$ is nice, then there exists a conservative family of fibre functors $\{\phi_i\}_{i\in I}$ such that $f$ is an equivalence if and only if for all $i\in I$, the induced morphism $\phi_i(f)$ is an equivalence. Does such a family also detect essential surjectivity, that is to say:

If for all $i\in I$, the $\phi_i(f)$ are essentially surjective, does it follow that for any $U\in \text{Ob}(\mathcal{C})$ the induced morphism $f(U):\text{Im}(f)(U)\to \mathcal{G}(U)$ is essentially surjective, where $\text{Im}(f)$ denotes the image $n$-sheaf? (I'm most interested in the $n=2$ case if that makes a difference).