  We prove that the derived parabolic induction functor, defined on the
unbounded derived category of smooth mod $p$ representations of a $p$-adic
reductive group, admits a left adjoint $\mathrm{L}(U,-)$. We study the
cohomology functors $\mathrm{H}^i\circ \mathrm{L}(U,-)$ in some detail and
deduce that $\mathrm{L}(U,-)$ preserves bounded complexes and global
admissibility in the sense of Schneider--Sorensen. Using $\mathrm{L}(U,-)$ we
define a derived Satake homomorphism und prove that it encodes the mod $p$
Satake homomorphisms defined explicitly by Herzig.
