Given a smooth manifold $M$ and a Lie algebroid $E\rightarrow M$ we can consider the $E$-polydifferential operators $D_E$ and the $E$-polyvectorfields $T_E$ as $$D_E:=\bigoplus_{k=-1}^\infty\mathcal{U}(E)^{\otimes k+1}\hspace{0.3cm}\mbox{and}\hspace{0.3cm}T_E=\bigoplus_{k=-1}^\infty\Gamma\left(M;\bigwedge^{k+1}E\right) $$ where $\mathcal{U}(E)$ denotes the universal enveloping Hopf algebroid of the Lie-Rinehart pair $(\Gamma(E),C^\infty(M))$ and the tensor products are over $C^\infty(M)$ on the left hand side. One now finds the usual DGLA structures given by the analogues of the Gerstenhaber bracket (expressed in terms of the coproduct) and the Hochschild differential (expressed as Gerstenhaber bracket with a certain degree 1 element) on the left and the Schouten-Nijenhuis beacket on the right. Calaque Showed in http://link.springer.com/article/10.1007/s00220-005-1350-5 that one can construct a Kontsevich type formality $L_\infty$-quasi-isomorphism $$T_E\longrightarrow D_E$$ by using the Fedosov type techniques developed by Dolgushev in his thesis.

Reccently I had been studying this situation and this made me wonder about the functorial properties of these maps. Namely, consider the functors $$T,D\colon LA(M)\longrightarrow L_\infty[W^{-1}]$$ from the category of Lie algebroids over $M$ to the category of $L_\infty$-algebras localized at the quasi-isomorphisms.

**Q: Is there a natural equivalence $T\rightarrow D$?**

The work by Calaque shows that certainly $T_E$ is isomorphic to $D_E$ for all $E\rightarrow M$, but, after searching and thinking for a while, I could not really obtain any results about this kind of functoriality of these isomorphisms.

One thing that seems necessary (as Stefan pointed out in the comments) is that we consider the category of pairs $(E\rightarrow M,\nabla_E)$ where $\nabla_E$ is a torsion-free $E$-connection, since such connections are essential in constructing Calaque's quasi-isomorphism. Morphisms in this category would then have to somehow preserve these connections though. This seems to warrant considering only inclusions (considering only isomorphisms doesn't serve any purpose it seems to me).

In case there is some "stupid" reason that this question either makes no sense or is not worth investigating I would also welcome an explanation of this stupid reason, of course.