Lemma 60.9.2. Assumptions as in Definition 60.8.1. The inclusion functor

\[ \text{Cris}(X/S) \to \text{CRIS}(X/S) \]

commutes with finite nonempty limits, is fully faithful, continuous, and cocontinuous. There are morphisms of topoi

\[ (X/S)_{\text{cris}} \xrightarrow {i} (X/S)_{\text{CRIS}} \xrightarrow {\pi } (X/S)_{\text{cris}} \]

whose composition is the identity and of which the first is induced by the inclusion functor. Moreover, $\pi _* = i^{-1}$.

**Proof.**
For the first assertion see Lemma 60.8.2. This gives us a morphism of topoi $i : (X/S)_{\text{cris}} \to (X/S)_{\text{CRIS}}$ and a left adjoint $i_!$ such that $i^{-1}i_! = i^{-1}i_* = \text{id}$, see Sites, Lemmas 7.21.5, 7.21.6, and 7.21.7. We claim that $i_!$ is exact. If this is true, then we can define $\pi $ by the rules $\pi ^{-1} = i_!$ and $\pi _* = i^{-1}$ and everything is clear. To prove the claim, note that we already know that $i_!$ is right exact and preserves fibre products (see references given). Hence it suffices to show that $i_! * = *$ where $*$ indicates the final object in the category of sheaves of sets. To see this it suffices to produce a set of objects $(U_ i, T_ i, \delta _ i)$, $i \in I$ of $\text{Cris}(X/S)$ such that

\[ \coprod \nolimits _{i \in I} h_{(U_ i, T_ i, \delta _ i)} \to * \]

is surjective in $(X/S)_{\text{CRIS}}$ (details omitted; hint: use that $\text{Cris}(X/S)$ has products and that the functor $\text{Cris}(X/S) \to \text{CRIS}(X/S)$ commutes with them). In the affine case this follows from Lemma 60.5.6. We omit the proof in general.
$\square$

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