  In this paper, we prove the categorical wall-crossing formula for certain
quivers containing the three loop quiver, which we call DT/PT quivers. These
quivers appear as Ext-quivers for the wall-crossing of DT/PT moduli spaces on
Calabi-Yau 3-folds. The resulting formula is a semiorthogonal decomposition
which involves quasi-BPS categories studied in our previous papers, and we
regard it as a categorical analogue of the numerical DT/PT correspondence. As
an application, we prove a categorical DT/PT correspondence for sheaves
supported on reduced plane curves in the affine three dimensional space.
