The Stacks project

Proof. Assume $\mathcal{Z} \subset \mathcal{Y}$ is a closed substack and $f \circ g$ factors through $\mathcal{Z}$. To prove the lemma it suffices to show that $f$ factors through $\mathcal{Z}$. Consider a scheme $T$ and a morphism $T \to \mathcal{X}$ given by an object $x$ of the fibre category of $\mathcal{X}$ over $T$. We will show that $f(x)$ is in fact in the fibre category of $\mathcal{Z}$ over $T$. Namely, the projection $T \times _\mathcal {X} \mathcal{W} \to T$ is a surjective, flat, locally finitely presented morphism. Hence there is an fppf covering $\{ T_ i \to T\}$ such that $T_ i \to T$ factors through $T \times _\mathcal {X} \mathcal{W} \to T$ for all $i$. Then $T_ i \to \mathcal{X}$ factors through $\mathcal{W}$ and hence $T_ i \to \mathcal{Y}$ factors through $\mathcal{Z}$. Thus $x|_{T_ i}$ is an object of $\mathcal{Z}$. Since $\mathcal{Z}$ is a strictly full substack, we conclude that $x$ is an object of $\mathcal{Z}$ as desired. $\square$