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How does the cohomology theory on de Rham (pre)stack compute de Rham cohomology?

Recently when I'm reading PTVV, Shifted Symplectic Structures, in Sec. 2.1 Mapping stacks, the authors use the identification $H^*(X,E)\simeq H^{*}_{dR}(Y/k,\mathcal{E})$ to show $X=Y_{dR}$ admits an ...
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