Let $C$ be a site and $p : F \mapsto C$ $p' : F' \mapsto C$ two categories fibred in groupoids over $C$ which are stacks (see, e.g., the definition here https://stacks.math.columbia.edu/tag/0268). For an object $X$, I will wright $F(X)$ the category whose objects $Y$ verify $p(Y) = X$ and whose arrows $f$ verify $p(f) = Id_{X}$. By definition of a fibred categories, if $f : X \mapsto X'$ is a morphism in $C$, then I can choose for each $x \in F(X)$ a pullback $f^{*}x$ by $f$ (and two such pullbacks are isomorph).
The problem : Let consider a morphism $g : F \mapsto F'$ of fibred categories. I suppose that for each object $X$ in $C$, the induced functor $F(X) \mapsto F'(X)$ is fully faithful. I also suppose for each $x $ in $F'(X)$, there exists a cover $\{f_{i} : X_{i} \mapsto X\}$ such that $f_{i}^{*}x \in F'(X_{i})$ is in the essential image of $g$.
I want to show $g$ is an isomorphism. I try a few things which give nothing. Do you have ideas?
Many thanks.
