Barbell twists are natural

For any oriented smooth $4$--manifold $X$ diffeomorphic to $(S^2\times D^2)^{\natural n}$ ($n\geq0$), the author establishes a natural isomorphism of abelian groups: $$\mathrm{Mod}(X,\partial X)\cong \mathrm{Mod}(D^4,\partial D^4)\times\wedge^2H_2(X;\mathbb{Z}),$$ concerning the (smooth) boundary-fi