Lemma 59.17.2. Let $D$ and $D(n)$ be as in (59.17.0.1) and (59.17.0.4). 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 59.5.4, and

in $\text{Cris}^\wedge (C/A)$.

Lemma 59.17.2. Let $D$ and $D(n)$ be as in (59.17.0.1) and (59.17.0.4). 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 59.5.4, and

\[ D(n) = \coprod \nolimits _{j = 0, \ldots , n} D \]

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

\[ \mathop{Mor}\nolimits _{\text{Cris}^\wedge (C/A)}(D, B) = \mathop{\mathrm{Hom}}\nolimits _ A((P, J), (B, \mathop{\mathrm{Ker}}(B \to C))) \]

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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