  Braverman, Finkelberg, and Nakajima define Kac-Moody affine Grassmannian
slices as Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories and
prove that their Coulomb branch construction agrees with the usual loop group
definition in finite ADE types. The Coulomb branch construction has good
algebraic properties, but its geometry is hard to understand in general.
  In finite types, an essential geometric feature is that slices embed into one
another. We show that these embeddings are compatible with the fundamental
monopole operators (FMOs), remarkable regular functions arising from the
Coulomb branch construction. Beyond finite type these embeddings were not
known, and our second result is to construct them for all symmetric Kac-Moody
types. We show that these embeddings respect Poisson structures under a mild
"goodness" hypothesis. These results give an affirmative answer to a question
posed by Finkelberg in his 2018 ICM address and demonstrate the utility of FMOs
in studying the geometry of Kac-Moody affine Grassmannian slices, even in
finite types.
