  Let $\Lambda=kQ/I$ be a Koszul algebra over a field $k$, where $Q$ is a
finite quiver. An algorithmic method for finding a minimal projective
resolution $\mathbb{F}$ of the graded simple modules over $\Lambda$ is given in
Green-Solberg. This resolution is shown to have a "comultiplicative" structure
in Green-Hartman-Marcos-Solberg, and this is used to find a minimal projective
resolution $\mathbb{P}$ of $\Lambda$ over the enveloping algebra $\Lambda^e$.
Using these results we show that the multiplication in the Hochschild
cohomology ring of $\L$ relative to the resolution $\mathbb{P}$ is given as a
cup product and also provide a description of this product. This
comultiplicative structure also yields the structure constants of the Koszul
dual of $\L$ with respect to a canonical basis over $k$ associated to the
resolution $\mathbb{F}$. The natural map from the Hochschild cohomology to the
Koszul dual of $\Lambda$ is shown to be surjective onto the graded centre of
the Koszul dual.
