Shifted Yangians are a family of algebras which can be used to quantize slices in the affine Grassmannian. They admit a family of coproducts which makes the direct sum of their category Os into a monoidal category. I will give a description of the Grothendieck ring of this monoidal category. In particular, I will explain that it is isomorphic to the Cox ring of an infinite Bott-Samelson variety.