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Remark 60.8.5 (Functoriality). Let $p$ be a prime number. Let $(S, \mathcal{I}, \gamma ) \to (S', \mathcal{I}', \gamma ')$ be a morphism of divided power schemes over $\mathbf{Z}_{(p)}$. Set $S_0 = V(\mathcal{I})$ and $S'_0 = V(\mathcal{I}')$. Let

\[ \xymatrix{ X \ar[r]_ f \ar[d] & Y \ar[d] \\ S_0 \ar[r] & S'_0 } \]

be a commutative diagram of morphisms of schemes and assume $p$ is locally nilpotent on $X$ and $Y$. Then we get a continuous and cocontinuous functor

\[ \text{CRIS}(X/S) \longrightarrow \text{CRIS}(Y/S') \]

by letting $(U, T, \delta )$ correspond to $(U, T, \delta )$ with $U \to X \to Y$ as the $S'$-morphism from $U$ to $Y$. Hence we get a morphism of topoi

\[ f_{\text{CRIS}} : (X/S)_{\text{CRIS}} \longrightarrow (Y/S')_{\text{CRIS}} \]

see Sites, Section 7.21.

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