The Stacks project

Remark 7.29.7. Notation and assumptions as in Lemma 7.29.6. If the site $\mathcal{D}$ has a final object and fibre products then the functor $u : \mathcal{D} \to \mathcal{C}'$ satisfies all the assumptions of Proposition 7.14.7. Namely, in addition to the properties mentioned in the lemma $u$ also transforms the final object of $\mathcal{D}$ into the final object of $\mathcal{C}'$. This is clear from the construction of $u$. Hence, if we first apply Lemmas 7.29.5 to $\mathcal{D}$ and then Lemma 7.29.6 to the resulting morphism of topoi $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D}')$ we obtain the following statement: Any morphism of topoi $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ fits into a commutative diagram

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]_ g \ar[r]_ f & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \ar[d]^ e \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \ar[r]^{f'} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}') } \]

where the following properties hold:

  1. the morphisms $e$ and $g$ are equivalences given by special cocontinuous functors $\mathcal{C} \to \mathcal{C}'$ and $\mathcal{D} \to \mathcal{D}'$,

  2. the sites $\mathcal{C}'$ and $\mathcal{D}'$ have fibre products, final objects and have subcanonical topologies,

  3. the morphism $f' : \mathcal{C}' \to \mathcal{D}'$ comes from a morphism of sites corresponding to a functor $u : \mathcal{D}' \to \mathcal{C}'$ to which Proposition 7.14.7 applies, and

  4. given any set of sheaves $\mathcal{F}_ i$ (resp. $\mathcal{G}_ j$) on $\mathcal{C}$ (resp. $\mathcal{D}$) we may assume each of these is a representable sheaf on $\mathcal{C}'$ (resp. $\mathcal{D}'$).

It is often useful to replace $\mathcal{C}$ and $\mathcal{D}$ by $\mathcal{C}'$ and $\mathcal{D}'$.

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