TITLE: Definition of forgetful functor
QUESTION [9 upvotes]: I was wondering if it is possible to make a formal definition of what it means for a functor to be forgetful. That is, using only the terminology of categories. I have seen so many examples of forgetful functors, but they have always been specific examples, and the fact that they were forgetful was "clear simply from what we untuitively mean by forgetful.
So, is there a formal definition?

REPLY [13 votes]: Here is a proof that every functor is "forgetful." Let $F: \mathcal{C} \to \mathcal{D}$ be a functor. Then $\mathcal{C}$ is equivalent (in fact, isomorphic) to the category of pairs $(x, y) \in \mathcal{C} \times \mathcal{D}$ such that $F(x) = y$, where morphisms are pairs $(f, F(f)): (x, y) \to (x', y')$. Under this equivalence, $F$ is the functor to $\mathcal{D}$ that "forgets $\mathcal{C}$." This seems like a silly construction, but it is simply the restriction of the projection functor $\mathcal{C} \times \mathcal{D} \to \mathcal{D}$ to the graph of $F$, and if we believe that the former functor is forgetful, then surely its restriction to a subcategory should be forgetful.
The adjective "forgetful" is only meaningful when it is placed into the context of some specified extra structure (or the various generalizations of "structure" suggested by Baez-Dolan-Bartels) that one is supposed to be forgetting. For instance, if the structure one cares about is "algebras over some arbitrary monad," then Beck's monadicity theorem gives necessary and sufficient conditions for a functor to be "forgetful" in this sense. Without specifying such a context, "forgetful" is meaningless.