A concrete category has only forgetful functors?

Question

I am confused about definition. If category $C$ is small and $F$ faithful functor such that $F:C\rightarrow SET$ where $F$ doesn't change the structure of $C$ then the pair $(C,F)$ is concrete category?

Answer 1

A concrete category is a category equipped with a faithful functor into $Set$. This usually can be thought of as a forgetful functor, since in cases like $Ab$ and $Ring$, the faithful functor $F$ just forgets the abelian group/ring structure. But the actual definition of a concrete category only involves the idea of a faithful functor.