Remark 60.9.3 (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 $\mathbf{Z}_{(p)}$. 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 $Y$. By analogy with Topologies, Lemma 34.3.17 we define
\[ f_{\text{cris}} : (X/S)_{\text{cris}} \longrightarrow (Y/S')_{\text{cris}} \]
by the formula $f_{\text{cris}} = \pi _ Y \circ f_{\text{CRIS}} \circ i_ X$ where $i_ X$ and $\pi _ Y$ are as in Lemma 60.9.2 for $X$ and $Y$ and where $f_{\text{CRIS}}$ is as in Remark 60.8.5.
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