Example of functor not full not faithfull

Question

I am trying to find a functor that is not full nor faithful, if anyone can give me any suggestion, I appreciate it.

Thanks

Answer 1

Looking at two groups $G,G'$ as categories with one object, functors $G\to G'$ identify with group homomorphisms. Any non injective non surjective group homomorphism will do then.

Answer 2

A constant functor (all objects map to the same object, all morphisms to the identity morphism) is usually neither full nor faithful.