A dual in the sense above might not be a dual object in other established senses of the word. For example,
The dual of a Banach space is generally not the dual in any sense that guarantees is left or right adjoint to , unless is finite-dimensional. (Here the monoidal product is the projective tensor product, which makes the closed category of Banach spaces into a closed monoidal category.) In particular, there is generally no appropriate unit of the form (where is the ground field).
For a compact closed category, a dual object (in one or another monoidal category sense; see category with duals) is the same as a dual in the closed category sense. But in general, monoidal duals are a stronger notion than duals in the closed category sense.
Of course, where the monoidal product is not braided or symmetric monoidal, due care must be exercised in distinguishing between between different types of monoidal dual (left or right dual).
If is a star-autonomous category, “dual object” has a different meaning: it usually is taken to mean where is the dualizing object. In a -autonomous category where and are not isomorphic (e.g., a Boolean algebra), this gives something genuinely different from “dual” in the sense of this page. But again, these two meanings, and , coincide in the case where the -autonomous category is compact closed.