  A handlebody-knot is a handlebody embedded in the 3-sphere. We establish a
uniform method to construct invariants for handlebody-links. We introduce the
category $\mathcal{T}$ of handlebody-tangles and present it by generators and
relations. The result tells us that every functor on $\mathcal{T}$ that gives
rise to invariants is derived from what we call a quantum-commutative
quantum-symmetric algebra in the target category. The example of such algebras
of our main concern is finite-dimensional unimodular Hopf algebras. We
investigate how those Hopf algebras give rise to handlebody-knot invariants.
