Lemma 8.8.4. Let $\mathcal{C}$ be a site. Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Z} \to \mathcal{Y}$ be morphisms of fibred categories over $\mathcal{C}$. In this case the stackification of the $2$-fibre product is the $2$-fibre product of the stackifications.

Proof. Let us denote $\mathcal{X}', \mathcal{Y}', \mathcal{Z}'$ the stackifications and $\mathcal{W}$ the stackification of $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$. By construction of $2$-fibre products there is a canonical $1$-morphism $\mathcal{X} \times _\mathcal {Y} \mathcal{Z} \to \mathcal{X}' \times _{\mathcal{Y}'} \mathcal{Z}'$. As the second $2$-fibre product is a stack (see Lemma 8.4.6) this $1$-morphism induces a $1$-morphism $h : \mathcal{W} \to \mathcal{X}' \times _{\mathcal{Y}'} \mathcal{Z}'$ by the universal property of stackification, see Lemma 8.8.2. Now $h$ is a morphism of stacks, and we may check that it is an equivalence using Lemmas 8.4.7 and 8.4.8.

Thus we first prove that $h$ induces isomorphisms of $\mathit{Mor}$-sheaves. Let $\xi , \xi '$ be objects of $\mathcal{W}$ over $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. We want to show that

$h : \mathit{Mor}(\xi , \xi ') \longrightarrow \mathit{Mor}(h(\xi ), h(\xi '))$

is an isomorphism. To do this we may work locally on $U$ (see Sites, Section 7.26). Hence by construction of $\mathcal{W}$ (see Lemma 8.8.1) we may assume that $\xi , \xi '$ actually come from objects $(x, z, \alpha )$ and $(x', z', \alpha ')$ of $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ over $U$. By the same lemma once more we see that in this case $\mathit{Mor}(\xi , \xi ')$ is the sheafification of

$V/U \longmapsto \mathop{\mathrm{Mor}}\nolimits _{\mathcal{X}_ V}(x|_ V, x'|_ V) \times _{\mathop{\mathrm{Mor}}\nolimits _{\mathcal{Y}_ V}(f(x)|_ V, f(x')|_ V)} \mathop{\mathrm{Mor}}\nolimits _{\mathcal{Z}_ V}(z|_ V, z'|_ V)$

and that $\mathit{Mor}(h(\xi ), h(\xi '))$ is equal to the fibre product

$\mathit{Mor}(i(x), i(x')) \times _{\mathit{Mor}(j(f(x)), j(f(x'))} \mathit{Mor}(k(z), k(z'))$

where $i : \mathcal{X} \to \mathcal{X}'$, $j : \mathcal{Y} \to \mathcal{Y}'$, and $k : \mathcal{Z} \to \mathcal{Z}'$ are the canonical functors. Thus the first displayed map of this paragraph is an isomorphism as sheafification is exact (and hence the sheafification of a fibre product of presheaves is the fibre product of the sheafifications).

Finally, we have to check that any object of $\mathcal{X}' \times _{\mathcal{Y}'} \mathcal{Z}'$ over $U$ is locally on $U$ in the essential image of $h$. Write such an object as a triple $(x', z', \alpha )$. Then $x'$ locally comes from an object of $\mathcal{X}$, $z'$ locally comes from an object of $\mathcal{Z}$, and having made suitable replacements for $x'$, $z'$ the morphism $\alpha$ of $\mathcal{Y}'_ U$ locally comes from a morphism of $\mathcal{Y}$. In other words, we have shown that any object of $\mathcal{X}' \times _{\mathcal{Y}'} \mathcal{Z}'$ over $U$ is locally on $U$ in the essential image of $\mathcal{X} \times _\mathcal {Y} \mathcal{Z} \to \mathcal{X}' \times _{\mathcal{Y}'} \mathcal{Z}'$, hence a fortiori it is locally in the essential image of $h$. $\square$

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