  Zagier introduced special bases for weakly holomorphic modular forms to give
the new proof of Borcherds' theorem on the infinite product expansions of
integer weight modular forms on $\SL_2(\ZZ)$ with a Heegner divisor. These good
bases appear in pairs, and they satisfy a striking duality, which is now called
the Zagier duality. After the result of Zagier, this type duality was studied
broadly in various view points including the theory of a mock modular form. In
this paper, we consider this problem with the Eichler cohomology theory,
especially the supplementary function theory developed by Knopp. Using
holomorphic Poincar\'e series and their supplementary functions, we construct a
pair of families of vector-valued harmonic weak Maass forms satisfying the
Zagier duality with integer weights $-k$ and $k+2$ respectively, $k>0$, for a
$H$-group. We also investigate the structures of them such as the images under
the differential operators $D^{k+1}$ and $\xi_{-k}$ and quadric relations of
the critical values of their $L$-functions.
