Given a completely distributive lattice L, we introduce two categories whose objects are L-powersets \(L^X\) of additive groups \((X,+,0)\) and the morphisms are respectively forward and backward operators \(R^\rightarrow: L^X \rightarrow L^Y\) and \(R^\leftarrow: L^Y \rightarrow L^X\) induced by L-relations \(R: X\times Y \rightarrow L\) which preserve the algebraic structure of the underlying groups \((X,+_X,0_X)\) and \((Y,+_Y,0_Y)\) . The objects of these categories serve as the field of action for operators of fuzzy erosion and dilation. We study some properties of these operators, in particular their behaviour under operators \(R^\rightarrow \) and  \(R^\leftarrow \) .