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

declarethat the effect of an endomorphism $f : X\to X$ over $S_0$ on the right sideisby 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.