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

ringedtopoi $$((X/W_n(k))_{\rm cris},\mathcal{O}_{X/W_n})\to ((Y/W_n(k))_{\rm cris},,\mathcal{O}_{Y/W_n})?$$ 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.