• A forgetful functor is a functor which is defined by ‘forgetting’ some structure.
• For example, the forgetful functor from $\mathbf{Grp}$ to $\mathbf{Set}$ forgets the group structure of a group, remembering only the underlying set.
• the term “forgetful functor” has no precise definition, being simply used whenever a functor is obviously defined by forgetting something.