The question is about Proposition 3.8.1 in Laumon and Moret-Bailly book on algebraic stacks.

Let $S$ be a scheme and let $F: \mathscr{X} \rightarrow \mathscr{Y}$ be a morphism of $S$-stacks (for the etale topology, say). Let $F': \mathscr{X}' \rightarrow \mathscr{Y}'$ be the base change of $F$ along an epimorphism of $S$-stacks $Q\colon \mathscr{Y}' \rightarrow \mathscr{Y}$. Suppose that $F'$ is a monomorphism. Is $F$ then necessarily a monomorphism, too?

This is claimed in the reference cited above, but the proof given there is unclear to me. In the proof they seem to use that $\mathscr{X}' \times_{\mathscr{Y}'} \mathscr{X}' \rightarrow \mathscr{X} \times_{\mathscr{Y}} \mathscr{X}$ is an epimorphism, but I cannot see why this is so, the subtlety being that these are $2$-fiber products, so objects take into account isomorphisms in $\mathscr{Y}'$ and $\mathscr{Y}$ and we cannot a priori lift an isomorphism in a fiber category of $\mathscr{Y}$ to $\mathscr{Y}'$.