  Recently, Oh and Thomas constructed algebraic virtual cycles for moduli
spaces of sheaves on Calabi-Yau 4-folds. The purpose of this paper is to
provide a virtual pullback formula between these Oh-Thomas virtual cycles. We
find a natural compatibility condition between 3-term symmetric obstruction
theories that induces a virtual pullback formula. There are two types of
applications.
  Firstly, we introduce a Lefschetz principle in Donaldson-Thomas theory, which
relates the tautological DT4 invariants of a Calabi-Yau 4-fold with the DT3
invariants of its divisor. As corollaries, we prove the Cao-Kool conjecture on
the tautological Hilbert scheme invariants for very ample line bundles and the
Cao-Kool-Monavari conjecture on the tautological DT/PT correspondence for line
bundles with Calabi-Yau divisors when the tautological complexes are vector
bundles.
  Secondly, we present a correspondence between the Oh-Thomas virtual cycles on
the moduli spaces of pairs and the moduli spaces of sheaves by combining the
virtual pullback formula and a pushforward formula for virtual projective
bundles. As corollaries, we prove the Cao-Maulik-Toda conjecture on the primary
PT/GV correspondence for irreducible curve classes and the Cao-Toda conjecture
on the primary JS/GV correspondence under the coprime condition, assuming the
Cao-Maulik-Toda conjecture on the primary Katz/GV correspondence. Moreover, we
also prove tautological versions of these two correspondences.
