  We construct several families of functors on the homotopy category of
singular Soergel bimodules that mimic cabling and insertion of column-colored
projectors. We use these functors to identify the intrinsically-colored
homology of Webster--Williamson and the projector-colored homology of
Elias--Hogancamp for an arbitrary link, up to multiplication by a polynomial in
the quantum degree $q$. Combined with the results of arXiv:2303.16271, this
establishes parity results for the intrinsic column-colored homology of
positive torus knots, partially resolving a conjecture of
Hogancamp--Rose--Wedrich in arXiv:2107.09590.
