  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 of
a Koszul algebra $A$, along with its cup product, is a subquotient of the
tensor product algebra $A^!\otimes A$ of $A$ with its Koszul dual.
