Choose a big $\mathit{fppf}$-site $(\mathbf{Sch})_{\mathit{fppf}}$ 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 algébriques, 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 $$\coprod_{n\in\mathbb{Z}}(\mathcal{X}_n\times_{\mathcal{Z}}\mathcal{Y})\cong \Big(\coprod_{n\in\mathbb{Z}}\mathcal{X}_n\Big)\times_{\mathcal{Z}}\mathcal{Y}\quad?$$

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

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 $$\coprod_{n\in\mathbb{Z}}(\mathcal{X}_n\times_{\mathcal{Z}}\mathcal{Y})\cong (\coprod_{n\in\mathbb{Z}}\mathcal{X}_n)\times_{\mathcal{Z}}\mathcal{Y}$$?

Choose a big $\mathit{fppf}$-site $(\mathbf{Sch})_{\mathit{fppf}}$ 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 algébriques, 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 $$\coprod_{n\in\mathbb{Z}}(\mathcal{X}_n\times_{\mathcal{Z}}\mathcal{Y})\cong \Big(\coprod_{n\in\mathbb{Z}}\mathcal{X}_n\Big)\times_{\mathcal{Z}}\mathcal{Y}\quad?$$