**Yoshifumi Tsuchimoto**

- (Sets)=(Category of all sets.) For Sets , the set of morphism is defined to be the set of all maps from to .
- (Groups)=(Category of all groups.) For Groups , the set of morphism is defined to be the set of all group homomorphisms from to .
- (Abelian groups), (Commutative Rings), are defined in a similar manner.
- For a given ring , we define ( -modules) to be the category of all -modules. Morphisms are -module homomorphisms.
- (Top)=(Category of all topological spaces.) For Top , the set of morphism is defined to be the set of all continuous maps from to .
- (Hausdorff Sp.)=(the category of all Hausdorff spaces),
(Compact Sp)=(the category of all Compact spaces)
are defined in a similar manner.

- An function which assigns to each object of an object of .
- An function which assigns to each morphism of an morphism of .

- functor-1.
- for any object of .
- functor-2.
- for any composable morphisms of .

By employing an axiom

(functor- ) for any composable morphisms

instead of axiom (functor-2) above, we obtain a definition of
a **contravariant functor**.