  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
$H(\mathbb{F}_{q})$.

Representation Theory;Algebraic Geometry