rt.representation theory - Is there a "categorical" description of Grothendieck's algebra of differential operators?

First, pick a commutative ring $k$ as the "ground field". Everything I say will be $k$-linear, e.g. "algebra" means "unital associative algebra over $k$". Then recall the following construction due to Grothendieck:

Let $A$ be a commutative algebra, and $X$ an $A$-module. Then the differential operators on $X$ is the filtered algebra $D = D(A,X)$ given inductively by:
$$ D_{leq 0} = text{image of $A$ in }hom_k(X,X) $$
$$ D_{leq n} = { phi in hom_k(X,X) text{ s.t. } [phi,a] in D_{leq n-1}\, forall ain A} $$
$$ D = bigcup_{n=0}^infty D_{leq n} $$
(Edit: In the comments, Michael suggests that $D_{leq 0} = hom_A(X,X)$ is the more standard definition, and the rest is the same.)

Then the following facts are more or less standard:

• If $A$ acts freely on $X$, then $D_{leq 1}$ acts on $A$ by derivations. (It's always true that $D_{leq 1}$ acts on $D_{leq 0}$ by derivations; the question is whether $D_{leq 0} = A$ or a quotient.) If $X = A$ by multiplication, then $D_{leq 1}$ splits as a direct sum $D_{leq 1} = text{Der}(A) oplus A$.


• If $k=mathbb R$, $M$ is a finite-dimensional smooth manifold, and $A = C^infty(M) = X$, then $D$ is the usual algebra of differential operators generated by $A$ and $text{Vect}(M) = Gamma(TM to M)$.


• If $A,X$ are actually sheaves, so is $D$.

Thus, at least in the situation where $A = X = C^{infty}(M)$, the algebra $D$ is acting very much like the universal enveloping algebra of $U (text{Vect}(M))$; in particular, the map $U(text{Vect}(M)) to D$ is filtered and is (almost) a surjection: it misses only the non-constant elements of $A$. So when $A = X = C^infty(-)$ are sheaves on $M$, it's very tempting to think of $D$ as a sheafy version of $U(text{Vect}(-))$. Note that $U(text{Vect}(-))$ is not a sheaf: its degree $leq 0$ part consists of constant functions, not locally constant, for example, and there are non-zero elements in $U_{leq 2}$ that restrict to $0$ on an open cover. I think that it cannot be true that the sheafification of $U(text{Vect}(-))$ is $D$, as the sheafification of $U_{leq 0}$ is the locally-constant sheaf, not $C^infty$.

So: is there a description of $D$ that makes it more obviously like a universal enveloping algebra? E.g. is there some adjunction or other categorical description? Is it really true that $D$ is a "sheafy" version of $U$ in a precise sense, or is this just a chimera?