Lemma 97.5.3. Let $S$ be a scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$ satisfy (RS), then so does $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$.

Proof. This is formal. Let

$\xymatrix{ X \ar[r] \ar[d] & X' \ar[d] \\ Y \ar[r] & Y' = Y \amalg _ X X' }$

be a diagram as in Definition 97.5.1. We have to show that

$(\mathcal{X} \times _{\mathcal{Y}} \mathcal{Z})_{Y'} \longrightarrow (\mathcal{X} \times _{\mathcal{Y}} \mathcal{Z})_ Y \times _{(\mathcal{X} \times _{\mathcal{Y}} \mathcal{Z})_ X} (\mathcal{X} \times _{\mathcal{Y}} \mathcal{Z})_{X'}$

is an equivalence. Using the definition of the $2$-fibre product this becomes

97.5.3.1
\begin{equation} \label{artin-equation-RS-fibre-product} \mathcal{X}_{Y'} \times _{\mathcal{Y}_{Y'}} \mathcal{Z}_{Y'} \longrightarrow (\mathcal{X}_ Y \times _{\mathcal{Y}_ Y} \mathcal{Z}_ Y) \times _{(\mathcal{X}_ X \times _{\mathcal{Y}_ X} \mathcal{Z}_ X)} (\mathcal{X}_{X'} \times _{\mathcal{Y}_{X'}} \mathcal{Z}_{X'}). \end{equation}

We are given that each of the functors

$\mathcal{X}_{Y'} \to \mathcal{X}_ Y \times _{\mathcal{Y}_ Y} \mathcal{Z}_ Y, \quad \mathcal{Y}_{Y'} \to \mathcal{X}_ X \times _{\mathcal{Y}_ X} \mathcal{Z}_ X, \quad \mathcal{Z}_{Y'} \to \mathcal{X}_{X'} \times _{\mathcal{Y}_{X'}} \mathcal{Z}_{X'}$

are equivalences. An object of the right hand side of (97.5.3.1) is a system

$((x_ Y, z_ Y, \phi _ Y), (x_{X'}, z_{X'}, \phi _{X'}), (\alpha , \beta )).$

Then $(x_ Y, x_{Y'}, \alpha )$ is isomorphic to the image of an object $x_{Y'}$ in $\mathcal{X}_{Y'}$ and $(z_ Y, z_{Y'}, \beta )$ is isomorphic to the image of an object $z_{Y'}$ of $\mathcal{Z}_{Y'}$. The pair of morphisms $(\phi _ Y, \phi _{X'})$ corresponds to a morphism $\psi$ between the images of $x_{Y'}$ and $z_{Y'}$ in $\mathcal{Y}_{Y'}$. Then $(x_{Y'}, z_{Y'}, \psi )$ is an object of the left hand side of (97.5.3.1) mapping to the given object of the right hand side. This proves that (97.5.3.1) is essentially surjective. We omit the proof that it is fully faithful. $\square$

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