  The classifying spaces of handlebody groups form a modular operad. Algebras
over the handlebody operad yield systems of representations of handlebody
groups that are compatible with gluing. We prove that algebras over the modular
operad of handlebodies with values in an arbitrary symmetric monoidal
bicategory $\mathcal{M}$ (we introduce for these the name ansular functor) are
equivalent to self-dual balanced braided algebras in $\mathcal{M}$. After
specialization to a linear framework, this proves that consistent systems of
handlebody group representations on finite-dimensional vector spaces are
equivalent to ribbon Grothendieck-Verdier categories in the sense of
Boyarchenko-Drinfeld. Additionally, it produces a concrete formula for the
vector space assigned to an arbitrary handlebody in terms of a generalization
of Lyubashenko's coend. Our main result can be used to obtain an ansular
functor from vertex operator algebras subject to mild finiteness conditions.
This includes examples of vertex operator algebras whose representation
category has a non-exact monoidal product.
