  Let $G$ be a $\mathbb{Q}_p$-split reductive group with connected centre and
Borel subgroup $B=TN$. We construct a right exact functor $D^\vee_\Delta$ from
the category of smooth modulo $p^n$ representations of $B$ to the category of
projective limits of finitely generated \'etale $(\varphi,\Gamma)$-modules over
a multivariable (indexed by the set of simple roots) commutative Laurent-series
ring. These correspond to representations of a direct power of
$\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ via an equivalence of
categories. Parabolic induction from a subgroup $P=L_PN_P$ corresponds to a
basechange from a Laurent-series ring in those variables with corresponding
simple roots contained in the Levi component $L_P$. $D^\vee_\Delta$ is exact
and yields finitely generated objects on the category $SP_A$ of finite length
representations with subquotients of principal series as Jordan-H\"older
factors. Lifting the functor $D^\vee_\Delta$ to all (noncommuting) variables
indexed by the positive roots allows us to construct a $G$-equivariant sheaf
$\mathfrak{Y}_{\pi,\Delta}$ on $G/B$ and a $G$-equivariant continuous map from
the Pontryagin dual $\pi^\vee$ of a smooth representation $\pi$ of $G$ to the
global sections $\mathfrak{Y}_{\pi,\Delta}(G/B)$. We deduce that
$D^\vee_\Delta$ is fully faithful on the full subcategory of $SP_A$ with
Jordan-H\"older factors isomorphic to irreducible principal series.
