Choose a big $fppf$-site $(\mathbf{Sch})_{fppf}$ and let $S$ be a scheme in that site. Let $\{\mathcal{X}_n\mid n\in \mathbb{Z}\}$ be a family of stacks in groupoids over $S$ and let $\mathcal{Y}\to\mathcal{Z}$ be a morphism of stacks in groupoids over $S$.
Let $\mathcal{X}\colon\!\!=\coprod_{n\in \mathbb{Z}}\mathcal{X}_n$ be the disjoint union as described in [Champs algebriques, G.Laumon/L.Moret-Bailly, (3.3)]. Let $-\times-$ denote the $2$-fibre product of stacks in groupoids over $S$. Is there a canonical Isomorphism of stacks $$\coprod_{n\in\mathbb{Z}}(\mathcal{X}_n\times_{\mathcal{Z}}\mathcal{Y})\cong (\coprod_{n\in\mathbb{Z}}\mathcal{X}_n)\times_{\mathcal{Z}}\mathcal{Y}$$?