In category theory, the concept of forgetful functor does not have a precise definition. I know of the following two candidate definitions, neither of which seems fully adequate.

In the formalism of stuff, structure, property, every functor is considered a forgetful functor. While this makes a lot of sense from the perspective of that framework, it doesn't match how the term "forgetful functor" is used in practice.

One could also try to define a forgetful functor as a right adjoint, based on the idea that a "free" functor is left adjoint to a forgetful functor. This also doesn't work, since then a product formation functor $(A, B) \mapsto A \times B$ would be forgetful while a coproduct formation functor $(A, B) \mapsto A + B$ would not be, and this tension is clearly undesirable.

In category theory, the concept of forgetful functor does not have a precise definition. I know of the following two candidate definitions, neither of which is fully adequate.

In the formalism of stuff, structure, property, every functor is considered a forgetful functor. While this makes a lot of sense from the perspective of that framework, it doesn't match how the term "forgetful functor" is used in practice.

One could also try to define a forgetful functor as a right adjoint, based on the idea that a "free" functor is left adjoint to a forgetful functor. This also doesn't work, since then a product formation functor $(A, B) \mapsto A \times B$ would be forgetful while a coproduct formation functor $(A, B) \mapsto A + B$ would not be, and this tension is clearly undesirable.