TITLE: Example of a forgetful functor that is not faithful.
QUESTION [7 upvotes]: In the forgetful functor Wikipedia article  I read that 

"[Forgetful] Functors that forget the extra sets need not be faithful; distinct morphisms respecting the structure of those extra sets may be indistinguishable on the underlying set."  

Can anyone give me an example of  a forgetful functor that is not faithful?

REPLY [7 votes]: If $\mathcal{C},\mathcal{D}$ are categories, then the projection functor $\mathcal{C} \times \mathcal{D} \to \mathcal{C}$ (which "forgets" the second coordinate) is not faithful (unless $\mathcal{D}$ is thin or $\mathcal{C}$ is empty).