# ct.category theory – What does a family of conservative fibre functors on \$n\$-topoi see?

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?

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