  We study the truncation functors and show the existence of projective cover
of each irreducible module in parabolic BGG category $\mathcal O$ over infinite
rank Lie algebra of types $\mathfrak{a,b,c,d}$. Moreover, $\mathcal O$ is a
Koszul category. As a consequence, the corresponding parabolic BGG category
$\overline{\mathcal O}$ over infinite rank Lie superalgebra of types
$\mathfrak{a,b,c,d}$ through the super duality is also a Koszul category.
