category with duals (list of them)
dualizable object (what they have)
n-category = (n,n)-category
n-groupoid = (n,0)-category
abstract duality: opposite category,
A dualizable object in a symmetric monoidal (∞,n)-category is called fully dualizable if the structure maps of the duality unit and counit each themselves have adjoints, which have adjoints, and so on.
In the symmetric monoidal (infinity,3)-category of monoidal categories and bimodule categories between them, the fully dualizable objects are (or at least contain) the fusion categories. See there for details.
The definition appears around claim 2.3.19 of