Lemma 97.3.3. Let $S$ be a locally Noetherian scheme. Let

$\xymatrix{ \mathcal{W} \ar[d] \ar[r] & \mathcal{Z} \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{Y} }$

be a $2$-fibre product of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $k$ be a finite type field over $S$ and $w_0$ an object of $\mathcal{W}$ over $k$. Let $x_0, z_0, y_0$ be the images of $w_0$ under the morphisms in the diagram. Then

$\xymatrix{ \mathcal{F}_{\mathcal{W}, k, w_0} \ar[d] \ar[r] & \mathcal{F}_{\mathcal{Z}, k, z_0} \ar[d] \\ \mathcal{F}_{\mathcal{X}, k, x_0} \ar[r] & \mathcal{F}_{\mathcal{Y}, k, y_0} }$

is a fibre product of predeformation categories.

Proof. This is a matter of unwinding the definitions. Details omitted. $\square$

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