We show that cosystolic expansion of sheaves on posets can be derived from local expansion conditions of the sheaf and the poset. When the poset at hand is a cell complex — typically a high dimensional expander — a sheaf may be thought of as generalizing coefficient groups used for defining homology and cohomology, by letting the coefficient group vary along the cell complex. Previous works established local criteria for cosystolic expansion only for simplicial complexes and with respect to constant coefficients. Our main technical contribution is providing a criterion that is more general in two ways: it applies to posets and sheaves, respectively.