  In this paper we consider the total space of the canonical bundle of P^2 and
we use a proposal by Hosono, together with results in Seidel and
Auroux-Katzarkov-Orlov, to deduce the right physical mirror equivalence between
D^b(K_{P^2}) and the derived Fukaya category of its mirror. By construction,
our equivalence is compatible with the mirror map between moduli spaces and
with the computation of Gromov--Witten invariants.
