Endoscopy for Hecke categories, character sheaves and representations

For a split reductive group $G$ over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group $H$ with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on $G$ with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group $H$, after passing to asymptotic versions. The other is a similar relationship between representations of $G(\mathbb{F}_q)$ with a fixed semisimple parameter and unipotent representations of $H(\mathbb{F}_{q})$.
Comment: This is a revised version of the paper with the same title published in Forum Math. Pi 8 (2020), e12. An error on a 3-cocycle is corrected and main statements are simplified. A Corrigendum follows the main article. 57 pages