  The geometric Satake equivalence and the Springer correspondence are closely
related when restricting to small representations of the Langlands dual group.
We prove this result for \'etale sheaves, including the case of the mixed
characteristic affine Grassmannian, assuming a sufficient ramification. In this
process, we construct a monoidal structure on the restriction functor of Satake
categories. We construct also a canonical isomorphism between a mixed
characteristic affine Grassmannian under a sufficient ramification and an equal
characteristic one.
