Faithful functors vs forgetful functors: exact category-theoretic defs?
you should have a look at http://ncatlab.org/nlab/show/forgetful+functor. As the nLab article says, in its common usage there is no precise definition of forgetful functor, though there are some typical examples (such as a functor $U : Alg \to Set$ forgetting some kind of algebraic structure, which will typically have a left adjoint $F : Set \to Alg$ corresponding to the construction of free algebras). And in another (useful) sense, every functor may be regarded as a forgetful functor, and classified according to how much it forgets.  There is no requirement that a functor be unfaithful for it to be called "forgetful", and many common examples of forgetful functors are faithful (such as the forgetful functor $Grp \to Set$ from groups to their underlying sets), while others are not (such as in general the functor $cod : C^\to \to C$ from the arrow category of $C$, forgetting everything about an arrow except its codomain).