The consistent systems of idempotents of Meyer and Solleveld allow to construct Serre subcategories of
Rep
R
(
G
)
\operatorname {Rep}_R(G)
, the category of smooth representations of a
p
p
-adic group
G
G
with coefficients in
R
R
. In particular, they were used to construct level 0 decompositions when
R
=
Z
¯
ℓ
R=\overline {\mathbb {Z}}_{\ell }
,
ℓ
≠
p
\ell \neq p
, by Dat for
G
L
n
GL_{n}
and the author for a more general group. Wang proved in the case of
G
L
n
GL_{n}
that the subcategory associated with a system of idempotents is equivalent to a category of coefficient systems on the Bruhat-Tits building. This result was used by Dat to prove an equivalence between an arbitrary level zero block of
G
L
n
GL_{n}
and a unipotent block of another group. In this paper, we generalize Wang’s equivalence of category to a connected reductive group on a non-archimedean local field.