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

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... quivers, which are expected to be conjectural Higgs branches of 3d Sicilian theories in type A by [BTX10]. 2 Note that if we instead only choose s and do not choose s then the resulting functor is represented by Gr X I × X I