We construct a braided monoidal functor J_4 from Bobtcheva and Piergallini's category 4\mathrm{HB} of connected relative 4-dimensional 2-handlebodies (up to 2-deformations) to an arbitrary unimodular ribbon category \mathcal{C}, which is not required to be semisimple. The main example of target category is provided by H-mod, the category of left modules over a unimodular ribbon Hopf algebra H. The source category 4\mathrm{HB} is generated, as a braided monoidal category, by a pre-modular Hopf algebra object, and this is sent by the Kerler-Lyubashenko functor J_4 to the end \int_{X \in \mathcal{C}} X \otimes X^* in \mathcal{C}, which is given by the adjoint representation in the case of H-mod. When \mathcal{C} is factorizable, we show that the construction only depends on the boundary and signature of relative handlebodies, and thus projects to a functor J_3^\sigma defined on Kerler's category 3\mathrm{Cob}^\sigma of connected framed relative 3-dimensional cobordisms. When H^* is not semisimple and H is not factorizable, our functor J_4 has the potential of detecting diffeomorphisms that are not 2-deformations.
