In a locally small category, the external hom (x, y) takes two objects to the set of morphisms. So in the category of sets, this is an object of the category itself. In the same vein, in a closed category, the morphisms from one object to another can be seen as lying inside the category. This is the internal hom [x,y].
and a fixed object I of V such that there is a natural isomorphism
and a dinatural transformation
- Cartesian closed categories are closed categories. In particular, any topos is closed. The canonical example is the category of sets.
- Compact closed categories are closed categories. The canonical example is the category FdVect with finite-dimensional vector spaces as objects and linear maps as morphisms.
- More generally, any monoidal closed category is a closed category. In this case, the object is the monoidal unit.