Ring objects in the equivariant derived Satake category arising from Coulomb branches

This is the second companion paper of [Part II]. We consider the morphism from the variety of triples introduced in [Part II] to the affine Grassmannian. The direct image of the dualizing complex is a ring object in the equivariant derived category on the affine Grassmannian (equivariant derived Satake category). We show that various constructions in [Part II] work for an arbitrary commutative ring object. The second purpose of this paper is to study Coulomb branches associated with star shaped quivers, which are expected to be conjectural Higgs branches of 3d Sicilian theories in type A by [BTX10].