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We prove that the spaces $\operatorname{tot}\big(\Gamma(\Lambda^\bulletA^\vee \otimes_R\mathcal{T}_{\operatorname{poly}}^{\bullet}\big)$ and$\operatorname{tot}\big(\Gamma(\Lambda^\bulletA^\vee)\otimes_R\mathcal{D}_{\operatorname{poly}}^{\bullet}\big)$ associatedwith a Lie pair $(L,A)$ each carry an $L_\infty$ algebra structure canonical upto an $L_\infty$ isomorphism with the identity map as linear part. These twospaces serve, respectively, as replacements for the spaces of formal polyvectorfields and formal polydifferential operators on the Lie pair $(L,A)$.Consequently, both$\mathbb{H}^\bullet_{\operatorname{CE}}(A,\mathcal{T}_{\operatorname{poly}}^{\bullet})$and$\mathbb{H}^\bullet_{\operatorname{CE}}(A,\mathcal{D}_{\operatorname{poly}}^{\bullet})$admit unique Gerstenhaber algebra structures. Our approach is based on homotopytransfer and the construction of a Fedosov dg Lie algebroid (i.e. a dgfoliation on a Fedosov dg manifold).

Polyvector fields and polydifferential operators associated with Lie pairs