  We construct an equivalence between the 2-categories VMonCat of rigid
V-monoidal categories for a braided monoidal category V and VModTens of oplax
braided functors from V into the Drinfeld centers of ordinary rigid monoidal
categories. The 1-cells in each are the respective lax monoidal functors, and
the 2-cells are the respective monoidal natural transformations. Our proof also
gives an equivalence in the case that we consider only strong monoidal 1-cells
on both sides. The 2-categories VMonCat and VModTens have G-graded analogues.
We also get an equivalence of 2-categories between G-extensions of some fixed
V-monoidal category A, and G-extensions of some fixed V-module tensor category
(A, F_A^Z).
