category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A monoidal fibration is a functor $\Phi\colon E\to B$ such that
$\Phi$ is a Grothendieck fibration (Cartesian fibration)
$E$ and $B$ are monoidal categories and $\Phi$ is a strict monoidal functor, and
the tensor product of $E$ preserves cartesian arrows.
If $B$ is cartesian monoidal, then monoidal fibrations over $B$ are equivalent to pseudofunctors $B^{op} \to MonCat$, which are called indexed monoidal categories. In this case the tensor product on $E$ is the external tensor product of the indexed monoidal category.