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We prove the $S=T$ conjecture proposed by Xiao-Zhu in\cite{2017arXiv170705700X}, making use of Scholze's theory of diamonds andv-stacks and Fargues-Scholze's geometric Satake equivalence. We deduce theEichler-Shimura relation for Shimura varieties of Hodge type.

S=T for Shimura Varieties and Moduli Spaces of p-adic Shtukas