Let $X$ be a complex algebraic variety. Let $C\subset X$ be a compact (reduced) subvariety. Let $C^{(n)}$ denote the $n$th infinitesimal neighborhood of $C$ inside $X$. Let $Hilb(X)$ denote the Hilbert scheme of $X$. Thus $C$ defines the point $[C]\in Hilb(X)$.

In my situation one may assume that $X$ and $C$ are smooth varieties, and moreover $[C]$ is a smooth point of $Hilb(X)$.

I was told (here Basic questions on the Hilbert scheme/ Douady space) that $Hilb(C^{(n)})$ is a closed subscheme of $Hilb(X)$.

**Question. Is it true that the $n$th infinitesimal neighborhood of the point $[C]$ in $Hilb(X)$ is equal to the connected component of $Hilb(C^{(n)})$ containing $[C]$ as a closed point?**

I am particularly interested in a modification of this question when $X,C$ are complex analytic rather than algebraic manifolds; in that case Hilbert scheme is replaced by Douady space.

References are most welcome.