What is the forgetful functor for this simple finite category (based on the cyclic group C3)?

Question

I am trying to understand what a forgetful functor is.

I have read the definition here but I still cannot figure out what the forgetful functor is exactly for the simple cyclic group C3 viewed as a finite category.

EDIT:

More precisely, let's consider the category with only on object: the cyclic group C3, and all the group homomorphisms from C3 to C3 as arrows, in this case, what is the forgetful functor?

Could someone please enlighten me ?

Answer 1

The forgetful functor from the category of groups to the category of sets simply forgets the group structure. Formally if your group is $C_3 = (\{0,1,2\},+)$ where the group operation $+$ is addition mod 3, then applying the forgetful functor to $C_3$ yields the set $\{0,1,2\}.$