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Let $k$ be a perfect field of characteristic $p>0$, $X$ a smooth projective $k$-variety.

Denote by $(X/W_n(k))_{\rm cris}$ the small crystalline site of $X$.

Let $f : X\to Y$ be a morphism of smooth projective $k$-varieties.

We know there is a morphism of (small) crystalline topoi

$$f_{\rm cris} = (f^{-1}_{\rm cris}, f_{\rm cris, *}) : (X/W_n(k))_{\rm cris}\to (Y/W_n(k))_{\rm cris}.$$

One can endow $(X/W_n(k))_{\rm cris}$ with a structure sheaf of rings $\mathcal{O}_{X/W_n}$ assigned by $(U,T,\delta)\mapsto \Gamma(T,\mathcal{O}_T)$.

Does $f$ induce a morphism of ringed topoi $$((X/W_n(k))_{\rm cris},\mathcal{O}_{X/W_n})\to ((Y/W_n(k))_{\rm cris},,\mathcal{O}_{Y/W_n})?$$ In other words is there a map of sheaves of rings on $(X/W_n(k))_{\rm cris}$: $$f^{-1}_{\rm cris}\mathcal{O}_{Y/W_n}\to \mathcal{O}_{X/W_n},$$ and what is a reference for this?

I know the answer to this is “yes”, but I don’t see it spelled out anywhere.

In Remark 5.14 in Berthelot-Ogus’ “Notes on crystalline cohomology”, he constructs a $f_{\rm cris}$ and proves this, but it is not clear to me that his construction agrees with the construction of $f_{\rm cris}$ given in the Stacks project, for instance.

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Let's first figure out why the definition given in Berthelot-Ogus coincides with the one from the Stacks project.

Unraveling the definition 5.8.3 we see that for a sheaf $G$ on $(Y/W)_{cris}$ the inverse image $f^{-1}G$ is the sheafification of the presheaf $$\varphi(G):(X\supset U\to T)\mapsto \mathrm{colim}_{T\to T'} G(T')$$ where the limit is taken over objects $(Y\supset U'\to T')$ with $f(U)\subset U'$ equipped with a map $T\to T'$ compatible with $f|_U:U\to U'$(this is not what's literally written there but rather they take the limit over maps between certain preshaves in 5.7.2, but those are just the same as the maps $T\to T'$).

The stacks project definition goes through the big Zariski site. Starting from a sheaf $G$ on $(Y/W)_{cris}$ we first get one on $(Y/W)_{CRIS}$ given by (the sheafification of) $(\pi^{-1}G)(Y\xleftarrow{s} Z\to T)=\mathrm{colim}_{T\to T'}G(T')$ where the limit is taken over thickenings $(X\supset U\to T')$ from the small site such that $s$ factors through $U$ and equipped with a map $T\to T'$ compatible with $s:Z\to U$. The pullback of a sheaf $F$ on $(Y/W)_{CRIS}$ to the big Zariski site of $X$ is given just by $(f_{CRIS}^{-1}F)(X\leftarrow Q\to T)=F(Y\xleftarrow{f}X\leftarrow Q\to T)$ and, finally, we get a sheaf on $(X/W)_{cris}$ we just by restricting to a subsite.

Combining all that, we get that the inverse image is the sheafification(here I'm using that instead of sheafifying after each step you can just sheafify once in the end) of the presheaf $$(X\supset U\to T)\mapsto \mathrm{colim}_{T\to T'}G(T')$$ where the colimit is taken over $(Y\supset U'\to T')$ such that the composition $U\to X\to Y$ factors through $U'$ equipped with a map $T\to T'$ satisfying the obvious compatibility. This is the same preasheaf as before! Thus, the pullback functors are the same(the morphisms of topoi are also the same because the direct image is the right adjoint of the inverse image).

Finally, the morphism $f^{-1}\mathcal{O}_Y\to \mathcal{O}_X $ comes from the map of presheaves $$ \varphi(\mathcal{O}_Y)\to \mathcal{O}_X $$ induced by $\mathcal{O}_Y(T')=\Gamma(T',\mathcal{O}_{T'})\to\Gamma(T,\mathcal{O}_T)=\mathcal{O}_X(T)$ and it extends to the sheafification because $\mathcal{O}_X$ is a sheaf.

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