  In this note, we generalize the linear duality between vector subbundles (or
equivalently quotient bundles) of dual vector bundles to coherent quotients $V
\twoheadrightarrow \mathscr{L}$ considered in arXiv:1811.12525, in the
framework of Kuznetsov's homological projective duality (HPD). As an
application, we obtain a generalized version of the fundamental theorem of HPD
for the $\mathbb{P}(\mathscr{L})$-sections and the respective dual sections of
a given HPD pair.
