Uniqueness and functoriality of Igusa stacks

Igusa stacks are p-adic geometric objects that roughly parametrize abelian varieties up to isogeny. In a joint work with Daniels, van Hoften, and Zhang, we constructed Igusa stacks for Hodge type Shimura data, and discussed how its cohomology relates to the cohomology of Shimura varieties. But there is another pleasant feature: Igusa stacks are unique if they exist, and automatically functorial with respect to morphisms of Shimura data. I will discuss this result as well as a crucial ingredient from rational p-adic Hodge theory, namely the rigidity of twistor realizations of de Rham local systems.