category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
The phrase cartesian functor has been used in category theory with several different (but related) meanings.
A functor between fibered categories has been called cartesian if it sends cartesian morphisms into cartesian morphisms.
A functor between categories with finite products has been called cartesian if it preserves finite products (that is, it is a strong monoidal functor between cartesian monoidal categories).
A functor between finitely complete categories has been called cartesian (notably in Sketches of an Elephant) if it preserves finite limits. This is the case just when the induced functor on codomain fibrations is cartesian in the first sense.
Because of this ambiguity, it is perhaps always better to use a more precise term such as “(strong) morphism of fibrations”, “cartesian monoidal functor” or “finite product preserving functor”, and “finitely continuous functor” or “lex functor”.
cartesian functor