  Let $G$ be the $F$-points of a connected reductive group over a
non-archimedean local field $F$ of residue characteristic $p$ and $R$ be a
commutative ring. Let $P=LU$ be a parabolic subgroup of $G$ and $Q$ be a
parabolic subgroup of $G$ containing $P$. We study the functor
$\mathrm{St}_Q^G$ taking a smooth $R$-representation $\sigma$ of $L$ which
extends to a representation $\mathrm{e}_G(\sigma)$ of $G$ trivial on $U$ to the
smooth $R$-representation $\mathrm{e}_G(\sigma) \otimes_R \mathrm{St}_Q^G(R)$
of $G$ where $\mathrm{St}_Q^G(R)$ is the generalised Steinberg representation.
