I'm interested in understanding the non-abelian Gauss-Manin connection on Carlos Simpson's relative de Rham Moduli space $M_{dR}(X/S,n)$ for a smooth projective morphism of schemes $X/S$. The scheme $M_{dR}(X/S,n)$ universally corepresents the functor $M^{\#}_{dR}(X/S,n)$ which assigns to an $S$-scheme $S'$ the set of isomorphism classes of vector bundles with integrable connections $(E,\nabla)$.
$B$ is any fixed base scheme, while $S$ is any scheme over $B$. $X$ is a scheme which is smooth and projective over $S$.
The ultimate Gauss-Manin connection is an isomorphism $p_1^{*}M_{dR}(X/S,n)\cong p_2^{*}M_{dR}(X/S,n)\ $compatible with natural cocycle conditions. This isomorphism should be over $(S \times_B S)^\wedge$ ie the formal completion of the diagonal, and the cocycle conditions should be over $(S \times_B S \times_B S)^\wedge$ as expected.
In chapter 8 of "Moduli of Representations of the Fundamental Group of a Smooth Projective Variety. II" Simpson explains this, but I do not understand his explanation. The text is unfortunately behind a paywall here http://link.springer.com/article/10.1007/BF02698895. I will outline the printed arguments. In lemmas 8.1 to 8.3 he proves an equivalence between such vector bundles with integrable connection $(E,\nabla)$ and crystals of vector bundles on $X/S$. In lemma 8.4 Simpson proves Grothendieck's lemma: that if $S_0 \subset S$ is a closed subscheme defined by a nilpotent ideal then there is a symbol equivalence of categories between crystals on $X/S$ and crystals on $X_0/S$ where $X_0=X \times S_0$ still considered as an $S$-scheme. He then quickly concludes the result. I think his main reasoning is on page 40.
On page 40, Simpson explains how it is advantageous to give an interpretation of vector bundles with integrable connection on $X/S$ as crystals. Indeed, if $S'$ is an $S$-scheme which contains a closed subscheme $S'_0$ defined by a nilpotent ideal and we set $X_0'=X' \times_{S'} S_0'$ then a cyrstal on $X_0'/S'$ is canonically the same thing as a crystal on $X_0'/S$. I follow so far. Then he says
"the set of crystals on $X_0'/S$ depends only on the restricted map $S_0' \to S$ so the functor $M^{\#}_{dR}(X/S,n)$ is itself a crystal on $S$." This seems to be using a version of Grothendieck where the second scheme is changing.
That is, he wants to conclude that the set of crystals on $X_0/S$ is the same as the set of crystals on $X_0/S_0$. I do not see how this follows.
In order to prove the existence of that connection, it is necessary to prove an isomorphism between the sets of crystals on $X_0/S$ and $X_0/S_0$... and it is sufficient to prove this in the case when $S$ has a retraction onto $S_0$. In this case, we can take a crystal on $X_0/S_0$ and "enlarge" it to $X_0/S$ by using a retraction to evaluate the crystal on pairs $(X \subset Y)$ over $X/S$ but this is not uniquely defined.
So how can I make $M^{\#}_{dR}(X/S,n)$ a crystal over the infinitesimal site of $S/B$ as I'm aiming to? Another very brief explanation by Simpson of this argument is found on page 5, theorem 12 of http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0747.0756.ocr.pdf. I feel Simpson's explanation lacks.... explanation and may be pulling the wool over my eyes. His explanation on how to change $X$ to $X_0$ is very detailed with almost no word on how to change $S$ to $S_0$.
