  We prove in this paper the weak consistency of a general finite volume
convection operator acting on discrete functions which are possibly not
piecewise-constant over the cells of the mesh and over the time steps. It
yields an extension of the Lax-Wendroff if-theorem for general colocated or
non-colocated schemes. This result is obtained for general polygonal or
polyhedral meshes, under assumptions which, for usual practical cases,
essentially boil down to a flux-consistency constraint; this latter is, up to
our knowledge, novel and compares the discrete flux at a face to the mean value
over the adjacent cell of the continuous flux function applied to the discrete
unknown function. We then apply this result to prove the consistency of a
finite volume discretisation of a convection operator featuring a (convected)
scalar variable and a (convecting) velocity field, with a staggered
approximation, i.e. with a cell-centred approximation of the scalar variable
and a face-centred approximation of the velocity.
