The Stacks project

Remark 60.8.7 (Structure morphism). In Situation 60.7.5. Consider the closed subscheme $S_0 = V(\mathcal{I}) \subset S$. If we assume that $p$ is locally nilpotent on $S_0$ (which is always the case in practice) then we obtain a situation as in Definition 60.8.1 with $S_0$ instead of $X$. Hence we get a site $\text{CRIS}(S_0/S)$. If $f : X \to S_0$ is the structure morphism of $X$ over $S$, then we get a commutative diagram of morphisms of ringed topoi

by Remark 60.8.5. We think of the composition $(X/S)_{\text{CRIS}} \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar})$ as the structure morphism of the big crystalline site. Even if $p$ is not locally nilpotent on $S_0$ the structure morphism

is defined as we can take the lower route through the diagram above. Thus it is the morphism of topoi corresponding to the cocontinuous functor $\text{CRIS}(X/S) \to (\mathit{Sch}/S)_{Zar}$ given by the rule $(U, T, \delta )/S \mapsto U/S$, see Sites, Section 7.21.