If I have a Deligne-Mumford stack $\Pi : X \to (\mathrm{Sch}/k)$ for some field $k$, can it be reconstructed from $\Pi^{-1}(C) \subset X$ for some "small" subcategory $C \subset (\mathrm{Sch}/k)$? For example, let's say we assume the fppf topology (to be concrete), then does the restriction to the fully faithful subcategory $T$ with objects

$\mathrm{Ob}(T) := $ { $I_n := \mathrm{Spec~} k[\epsilon]/(\epsilon^n) \mid n \in \mathbf{N}$ }

completely determine the stack? I think I've read that the restriction to $(\mathrm{Aff}/k)$ is enough, but since the fibers over the $I_n$ determine the $n^{th}$-order formal neighborhoods ($\mathrm{HOM}(I_n, X) \cong X(I_n)$), I wonder if the restriction to $T$ is enough?