category with duals (list of them)
dualizable object (what they have)
An object in a symmetric monoidal -category is called dualizable if …
Let be a symmetric monoidal -category. Then there exists another symmetric monoidal -category and a symmetric monoidal functor
such that has duals and is universal with these properties:
for any symmetric monoidal -category with duals and any symmetric monoidal functor there exists a symmetric monoidal functor , unique up to equivalence, and an equivalence
This appears as (Lurie, claim 2.3.19).
is obtained from by discarding all objects that do not have duals and all k-morphisms that do not admit right and left adjoints.
An object is called fully dualizable if it is in the essential image of .
A discussion of dualizable objects is in section 2.3 of