If is a closed category (for example a closed monoidal category), with the internal hom denoted as and the unit object as , then the dual of an object is usually taken to mean . This applies for example to categories of modules over a commutative ring, to the category of Banach spaces, etc., with their usual closed monoidal structures.
A dual in the sense above might not be a dual object in other established senses of the word. For example,
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).