category with duals (list of them)
dualizable object (what they have)
The precise definition associated with the term “tensor category” varies somewhat in the literature.
It may mean:
any monoidal category,
an additive (symmetric) monoidal category, so that the tensor preserves finite direct sums,
an abelian (symmetric) monoidal category, in which the tensor preserves finite colimits in separate arguments,
an abelian (symmetric) monoidal category with dual objects.
Discussion with an eye towards superalgebra is in