  Let $X$ be a Calabi-Yau 4-fold and $D$ a smooth connected divisor on it. We
consider tautological bundles of $L=\mathcal{O}_X(D)$ on Hilbert schemes of
points on $X$ and their counting invariants defined by integrating the Euler
classes against the virtual classes. We relate these invariants to
Maulik-Nekrasov-Okounkov-Pandharipande's invariants of Hilbert schemes of
points on $D$ by virtual pullback technique and confirm a conjecture of
Cao-Kool. The same strategy is also applied to obtain a virtual pullback
formula for tautological invariants of one dimensional stable sheaves. This in
turn gives a nontrivial identity on primary and descendent invariants as
studied in the works of Cao, Maulik and Toda.
