Lemma 60.17.2. Let $D$ and $D(n)$ be as in (60.17.0.1) and (60.17.0.4). Then $(D, \bar J, \bar\gamma )$ and $(D(n), \bar J(n), \bar\gamma (n))$ are objects of $\text{Cris}^\wedge (C/A)$, see Remark 60.5.4, and
in $\text{Cris}^\wedge (C/A)$.
Lemma 60.17.2. Let $D$ and $D(n)$ be as in (60.17.0.1) and (60.17.0.4). Then $(D, \bar J, \bar\gamma )$ and $(D(n), \bar J(n), \bar\gamma (n))$ are objects of $\text{Cris}^\wedge (C/A)$, see Remark 60.5.4, and
in $\text{Cris}^\wedge (C/A)$.
Proof. The first assertion is clear. For the second, if $(B \to C, \delta )$ is an object of $\text{Cris}^\wedge (C/A)$, then we have
and similarly for $D(n)$ replacing $(P, J)$ by $(P \otimes _ A \ldots \otimes _ A P, J(n))$. The property on coproducts follows as $P \otimes _ A \ldots \otimes _ A P$ is a coproduct. $\square$
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