category with duals (list of them)
dualizable object (what they have)
The phrase cartesian functor has been used in category theory with several different (but related) meanings.
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”.