Lemma 18.7.2. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. There exists a factorization

$\xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \ar[rr]_{(f, f^\sharp )} \ar[d]_{(g, g^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) \ar[d]^{(e, e^\sharp )} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[rr]^{(h, h^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) }$

where

1. $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$ is an equivalence of topoi induced by a special cocontinuous functor $\mathcal{C} \to \mathcal{C}'$ (see Sites, Definition 7.29.2),

2. $e : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D}')$ is an equivalence of topoi induced by a special cocontinuous functor $\mathcal{D} \to \mathcal{D}'$ (see Sites, Definition 7.29.2),

3. $\mathcal{O}_{\mathcal{C}'} = g_*\mathcal{O}_\mathcal {C}$ and $g^\sharp$ is the obvious map,

4. $\mathcal{O}_{\mathcal{D}'} = e_*\mathcal{O}_\mathcal {D}$ and $e^\sharp$ is the obvious map,

5. the sites $\mathcal{C}'$ and $\mathcal{D}'$ have final objects and fibre products (i.e., all finite limits),

6. $h$ is a morphism of sites induced by a continuous functor $u : \mathcal{D}' \to \mathcal{C}'$ which commutes with all finite limits (i.e., it satisfies the assumptions of Sites, Proposition 7.14.7), and

7. 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. $\mathcal{D}'$).

Moreover, if $(f, f^\sharp )$ is an equivalence of ringed topoi, then we can choose the diagram such that $\mathcal{C}' = \mathcal{D}'$, $\mathcal{O}_{\mathcal{C}'} = \mathcal{O}_{\mathcal{D}'}$ and $(h, h^\sharp )$ is the identity.

Proof. This follows from Sites, Lemma 7.29.6, and Sites, Remarks 7.29.7 and 7.29.8. You just have to carry along the sheaves of rings. Some details omitted. $\square$

There are also:

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