  Let $o$ be the ring of integers in a finite extension $K/\mathbb{Q}_p$ and
$G=\mathbf{G}(\mathbb{Q}_p)$ be the $\mathbb{Q}_p$-points of a
$\mathbb{Q}_p$-split reductive group $\mathbf{G}$ defined over $\mathbb{Z}_p$
with connected centre and split Borel $\mathbf{B}=\mathbf{TN}$. We show that
Breuil's pseudocompact $(\varphi,\Gamma)$-module $D^\vee_{\xi}(\pi)$ attached
to a smooth $o$-torsion representation $\pi$ of $B=\mathbf{B}(\mathbb{Q}_p)$ is
isomorphic to the pseudocompact completion of the basechange
$\mathcal{O_E}\otimes_{\Lambda(N_0),\ell}\widetilde{D_{SV}}(\pi)$ to Fontaine's
ring (via a Whittaker functional $\ell\colon N_0=\mathbf{N}(\mathbb{Z}_p)\to
\mathbb{Z}_p$) of the \'etale hull $\widetilde{D_{SV}}(\pi)$ of $D_{SV}(\pi)$
defined by Schneider and Vigneras. Moreover, we construct a $G$-equivariant map
from the Pontryagin dual $\pi^\vee$ to the global sections $\mathfrak{Y}(G/B)$
of the $G$-equivariant sheaf $\mathfrak{Y}$ on $G/B$ attached to a
noncommutative multivariable version $D^\vee_{\xi,\ell,\infty}(\pi)$ of
Breuil's $D^\vee_{\xi}(\pi)$ whenever $\pi$ comes as the restriction to $B$ of
a smooth, admissible representation of $G$ of finite length.
