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Given an acyclic twisting cochain $\pi:C\to A$, from a curved dg coalgebra$C$ to a dg algebra $A$, we show that the associated twisted hom complex$\mathrm{Hom}^\pi_k(C,A)$ has cohomology equal to the Hochschild cohomology of$A$, as a graded ring. As a corollary we find that the Hochschild cohomology ofa Koszul algebra $A$, along with its cup product, is a subquotient of thetensor product algebra $A^!\otimes A$ of $A$ with its Koszul dual.

The cup product on Hochschild cohomology via twisting cochains and applications to Koszul rings