In a functor category the object is a functor.
|Morphisms are structure preserving maps between these functors.|
|Since the objects are all functors from X to Y we can redraw it like this:|
So we can the that these morphisms are really natural transformations (as described on page here).
So a functor category is a category where:
- The objects are functors.
- The morphisms are natural transformations.
Example of a Functor Category
See presheaves on page here.