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 ${phi_i}_{iin I}$ such that $f$ is an isomorphism if and only if for all $iin I$, 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 $iin I$ the $phi_i(f)$ are natural equivalences, does it follow that for any $Uin text{Ob}(mathcal{C})$ 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 $iin I$, the $phi_i(f)$ are essentially surjective, does it follow that for any $Uin text{Ob}(mathcal{C})$ 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?