ABSTRACT
Functors being the functions of the collection of arrows of a category, their composition also preserves identities as well as the composition of arrows. Hence, functors can be composed to yield another functor. The object side of a functor is, nevertheless, completely determined by the arrows side for a functor, for identity arrows uniquely label objects of a category. Consequently, the action of a functor on a hom-collection of a category can be expected to play an important role in determining the properties of a functor. A functor is an isoarrow iff it is full, faithful, as well as bijective on objects. Then an isomorphism of categories is seen to be a very restrictive concept about the relationship of the collections of arrows and objects in two categories connected by a functor. The cardinality of the collection of arrows of a category is not preserved by an equivalence functor. Natural transformations therefore connect functors having the same source and target.
