The Stacks project

Situation 91.12.2. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}), \mathcal{O}_\mathcal {B})$ be a morphism of ringed topoi. Consider a commutative diagram

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_1), \mathcal{O}'_1) \ar[r]_ h \ar[d]_{f'_1} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_2), \mathcal{O}'_2) \ar[d]_{f'_2} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}'_1), \mathcal{O}_{\mathcal{B}'_1}) \ar[r] & (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}'_2), \mathcal{O}_{\mathcal{B}'_2}) } \]

where $h$ is a morphism of first order thickenings of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$, the lower horizontal arrow is a morphism of first order thickenings of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{B}), \mathcal{O}_\mathcal {B})$, each $f'_ i$ restricts to $f$, both pairs $(f, f_ i')$ are strict morphisms of thickenings, and both $f'_ i$ are flat. Finally, let $\mathcal{F}$ be an $\mathcal{O}$-module flat over $\mathcal{O}_\mathcal {B}$.

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