  We denote the monoidal bicategory of two-sided modules (also called
profunctors, bimodules and distributors) between categories by $\mathrm{Mod}$;
the tensor product is cartesian product of categories. For a groupoid
$\scr{G}$, we study the monoidal centre
$\mathrm{ZPs}(\scr{G},\mathrm{Mod}^{\mathrm{op}})$ of the monoidal bicategory
$\mathrm{Ps}(\scr{G},\mathrm{Mod}^{\mathrm{op}})$ of pseudofunctors and
pseudonatural transformations; the tensor product is pointwise. Alexei Davydov
defined the full centre of a monoid in a monoidal category. We define a higher
dimensional version: the full monoidal centre of a monoidale (= pseudomonoid)
in a monoidal bicategory $\scr{M}$, and it is a braided monoidale in the
monoidal centre $\mathrm{Z}\scr{M}$ of $\scr{M}$. Each fibration $\pi : \scr{H}
\to \scr{G}$ between groupoids provides an example of a full monoidal centre of
a monoidale in $\mathrm{Ps}(\scr{G},\mathrm{Mod}^{\mathrm{op}})$. For a group
$G$, we explain how the $G$-graded categorical structures, as considered by
Turaev and Virelizier in order to construct topological invariants, fit into
this monoidal bicategory context. We see that their structures are monoidales
in the monoidal centre of the monoidal bicategory of $k$-linear categories on
which $G$ acts.
