In this series of two talk I will describe some older and some recent constructions and results about the geometry of stacks of equivariant ˇG-bundles on the Fargues-Fontaine curve. These stacks (with ˇG being the dual group of a fixed reductive group over $\mathbb{Q}_{p}$) replace the stacks of L-parameters in a categorical approach to a locally analytic p-adic Langlands correspondence. I will describe some expected features of this correspondence, focussing on the counterpart on the Galois side of some ”additional Hecke operators“ in the locally analytic world and on the relation with (geometric) properties of eigenvarieties.