Let $S$ be a scheme and let $\require{AMScd}$ \begin{CD} X' @>{p}>> X\\ @V{f'}VV @V{f}VV\\ Y' @>{q}>> Y \end{CD} be a Cartesian diagram in the category of sheaves on $\mathscr{S}ch/S$ with the etale topology. Assume that $q$ is epi. If $f'$ is representable by quasi-separated algebraic spaces, then so is $f$.
(For a morphism of sheaves $f : X \to Y$ over $S$, we say that $f$ is representable by quasi-separated algebraic spaces if for any $S$-scheme $T$ and any map $T \to Y$, $X \times_Y T$ is a quasi-separated algebraic space.)
To show it, it suffices to show that $X$ is a quasi-separated algebraic space, assuming that $Y, Y'$ affine schemes, $q$ etale surjective, and $X'$ a quasi-separated algebraic space.
And by 1.6. of Laumon, Moret-Bailly's Champs algébriques, to see that $X$ is a quasi-separated algebraic space, it suffices to show that $X' \times_X X' \to X' \times_S X'$ is quasi-compact.
But I can't show it.
Since $X' \times_S X' \to X \times_S X$ is representable by schemes, and is etale surjective, the quasi-compactness of $X' \times_X X' \to X' \times_S X'$ and the one of $X \to X \times_S X$ are equivalent. And of course in general $X \to X \times_S X$ is not quasi-compact. So it seems that this highlighted statement is wrong.
Where is wrong in my argument?