  We present an easily applicable sufficient condition for standard Koszul
algebras to be Koszul with respect to $\Delta$. If a quasi-hereditary algebra
$\L$ is Koszul with respect to $\Delta$, then $\L$ and the Yoneda extension
algebra of $\Delta$ are Koszul dual in a sense explained below, implying in
particular that their bounded derived categories of finitely generated graded
modules are equivalent. We also prove that the extension algebra of $\Delta$ is
Koszul in the classical sense.
