Let $A$ be an associative algebra, then we have the Hochschild chain complex, namely $$ \dotsb \to A^{\otimes 3} \xrightarrow{d_2} A^{\otimes 2} \xrightarrow{d_1} A \,, $$ where, for example, $d_1 (a \otimes b) = ab - ba.$ I want to apply $\operatorname{Hom}(-, A)$ to this complex. Unfortunately, after applying I obtain a complex with a differential that differs from the Hochschild differential.
Is it possible to obtain the Hochschild cochain complex from the Hochschild chain complex by applying something like $\operatorname{Hom}(-, A)$?
If $A$ is commutative, then we have an identification between the Hochschild cohomology and the polyvector fields $HH^i(A, A) \cong \Lambda^i \operatorname{Der}(A)$, suppose that $A$ is Poisson, then we have the Lichnerowicz–Poisson differential on polyvector fields. How to write it down on the level of the Hochschild cohomology (not only for commutative algebras)? (Actually, it'd interesting even for symplectic manifold for which we can identify polyvector fields and differential forms and then the question will be about the de-Rham differential on the level of Hochschild cohomology).