# Berthelot’s comparison theorem and functoriality

Let $$A$$ be a noetherian $$p$$-adically complete ring with an ideal $$I$$ equipped with a PD structure and such that $$p$$ is nilpotent on $$A/I$$.

Let $$S = \text{Spec}(A)$$, $$S_0 = \text{Spec}(A/I)$$, $$Y\to S$$ a smooth and proper morphism, $$X := Y\times_SS_0$$.

Berthelot’s comparison Theorem in crystalline cohomology says that we have:

$$R\Gamma(\text{Cris}(X/S),\mathcal{F}) = R\Gamma(Y,\mathcal{F}_Y\otimes_{\mathcal{O}_Y}\Omega^*_{Y/S})$$

where $$\Omega_{Y/S}^*$$ is the usual algebraic de Rham complex of $$Y/S$$, $$\mathcal{F}$$ is a finite type crystal in quasi-coherent $$\mathcal{O}_{X/S}$$-modules, $$\mathcal{F}_Y$$ is given by formal GAGA after restricting $$\mathcal{F}$$ to $$Y\times_S\text{Spec}(A/I^n)$$ for every $$n\ge 1$$ and algebrizing.

In the literature usually one then says “and then one sees that the right side is functorial in $$X$$”.

This sounds to me like a bit of hand waving. The right side is functorial in $$Y$$ and who knows if endomorphisms of $$X$$ over $$S_0$$ lift to endomorphisms of $$Y$$ over $$S$$.

Here is the question:

Does this sentence actually mean that one can declare that the effect of an endomorphism $$f : X\to X$$ over $$S_0$$ on the right side is by decree the effect on the left side, pre and post-composed with the above canonical comparison map?

For background on this comparison isomorphism see the Stacks Project or Berthelot-Ogus.