Formality theorem with coefficients in a module

In this article, $X$ will denote a ${\mathcal{C}}^{\mathrm{\infty }}$ manifold. In a very famous article, Kontsevich showed that the differential graded Lie algebra (DGLA) of polydifferential operators on $X$ is formal. Calaque extended this theorem to any Lie algebroid. More precisely, given any Lie algebroid $E$ over $X$, he defined the DGLA of $E$-polydifferential operators, $\Gamma (X, ^{E}D^{}_{poly})$,andshowedthatitisformal\mathrm{.}Denoteby$\Gamma (X, ^{E}T^{}{poly})$theDGLAof$E$-polyvectorfields\mathrm{.}Considering$M$, a module over $E$, we define $\mathrm{\Gamma }(X{,}^E{T}_{poly}^{\ast }(M))$ the $\Gamma (X, ^{E}T^{*}{poly})$-moduleof$E$-polyvectorfieldswithvaluesin$M$. Similarly, we define the $\mathrm{\Gamma }(X{,}^E{D}_{poly}^{\ast })$-module of $E$-polydifferential operators with values in $M$, $\Gamma (X, ^{E}D^{}{poly}(M))$\mathrm{.}Weshowthatthereisaquasi-isomorphismof$L{\infty}$-modulesover$\Gamma (X, ^{E}T^{}{poly})$from$\Gamma (X, ^{E}T^{*}{poly}(M))$to$\Gamma (X, ^{E}D^{*}_{poly}(M))$. Our result extends Calaque 's (and Kontsevich’s) result.