  On a smooth projective threefold, we construct an essentially surjective
functor $\mathcal{F}$ from a category of two-term complexes to a category of
quotients of coherent sheaves, and describe the fibers of this functor. Under a
coprime assumption on rank and degree, the domain of $\mathcal{F}$ coincides
with the category of higher-rank PT stable objects, which appear on one side of
Toda's higher-rank DT/PT correspondence formula. The codomain of $\mathcal{F}$
is the category of objects that appear on one side of another correspondence
formula by Gholampour-Kool, between the generating series of topological Euler
characteristics of two types of quot schemes.
