Let $S$ be a scheme, $X/S$ a "nice" scheme (I think "nice" = separated) and $e : S \to X$ a section. Let $\hat{X}$ be the formal completion of $X$ along the section $e$. (i.e., = $\lim_n S_n$, where $S_n$ is the "$n$-th-thickened of $S \to X$".) Then does $\hat{X}$ represent (or pro-represent) a nice functor?

First, the case that $S$ is a field $k$:
Let $\mathscr{C}_k$ be a category of artin local $k$-algebras with the residue field $k$. (Maps are local $k$-homomorphisms.) and $\hat{\mathscr{C}}_k$ be a category of complete local noetherian $k$-algebra with the residue field $k$.
And consider a functor $$\mathscr{C}_k \to \mathscr{S}et \\ A \mapsto \{ f : \operatorname{Spec}A \to X, \text{ an $S$-morphism which takes the only closed point to $e$ } \}. $$

Then this is $\operatorname{Hom}_{\mathscr{C}_k}(\mathscr{O}_{X,e}, A) = \operatorname{Hom}_{\hat{\mathscr{C}}_k}( \hat{\mathscr{O}_{X,e}}, A) = \hat{X}(A).$ So this functor is pro-represented by $\hat{X}$.
And if $X$ is a group scheme, then this functor is $\operatorname{ker} (X(A) \to X(A/\mathfrak{m}_A))$, so inherits the group structure of $X$.

How about the general case? Here is what I have tried:
Let $\mathscr{C}_S$ be the full subcategory of $\mathscr{S}ch/S$ whose objects are the spectrum of artin local ring. Then by the same arguments, it seems that the functor $$ \mathscr{C}_S^{\text{op}} \to \mathscr{S}et \\ T \mapsto \{ f : T \to X, \text{ an $S$-morphism whose set-theoritically image is contained in $e(S)$} \}$$

is equals to $\hat{X}(T)$. However $\hat{X}$ is not in " $\hat{\mathscr{C}}_S$ ", so this functor is "bad" in order to study $\hat{X}$.

If $S$ is affine and if we take $\mathscr{C}_S$ to be the category of finite $S$-schemes of finite length, then it seems to $\hat{X} \in \hat{\mathscr{C}_S}$. But I cannot show that $\hat{X}$ pro-represents that functor, and this does not coincide with the case that $S$ is a field.

Thank you very much!


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