The Stacks project

Remark 60.9.7 (Compatibilities). The morphisms defined above satisfy numerous compatibilities. For example, in the situation of Remark 60.9.3 we obtain a commutative diagram of ringed topoi

\[ \xymatrix{ (X/S)_{\text{cris}} \ar[d] \ar[r] & (Y/S')_{\text{cris}} \ar[d] \\ \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar}) \ar[r] & \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S')_{Zar}) } \]

where the vertical arrows are the structure morphisms.

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