  Assume the existence of a Fukaya category $\mathrm{Fuk}(X)$ of a compact
symplectic manifold $X$ with some expected properties. In this paper, we show
$\mathscr{A} \subset \mathrm{Fuk}(X)$ split generates a summand
$\mathrm{Fuk}(X)_e \subset \mathrm{Fuk}(X)$ corresponding to an idempotent $e
\in QH^\bullet(X)$ if the Mukai pairing of $\mathscr{A}$ is perfect. Moreover
we show $HH^\bullet(\mathscr{A}) \cong QH^\bullet(X) e$. As an application we
compute the quantum cohomology and the Fukaya category of a blow-up of
$\mathbb{C} P^2$ at four points with a monotone symplectic structure.
