According to "Champs algébriques" by Laumon and Moret-Bailley, and $S$-groupoid is a category $\mathscr{X}$ and a functor $a: \mathscr{X} \to (\mathrm{Aff}/S)$, where $(\mathrm{Aff}/S)$ is the category of affine schemes over a fixed scheme $S$. Moreover, it satisfies the properties that make $\mathscr{X}$ a fibered category over $(\mathrm{Aff}/S)$ where each fiber category $\mathscr{X}_U$ is a groupoid.
Then the authors assert that a morphism $F: \mathscr{X} \to \mathscr{Y}$ is a monomorphism if each restriction to the fiber categories $F_U: \mathscr{X}_U \to \mathscr{Y}_U$ is fully faithful. My question is the following: why is it not sufficient that each $F_U$ be only faithful? Why is fullness necessary?