forgetful functor

Let $\mathcal{C}$ and $\mathcal{D}$ be categories such that each object $c$ of $\mathcal{C}$ can be regarded an object of $\mathcal{D}$ by suitably ignoring structures $c$ may have as a $\mathcal{C}$ -object but not a $\mathcal{D}$ -object. A functor $U:\mathcal{C} \to \mathcal{D}$ which operates on objects of $\mathcal{C}$ by ``forgetting'' any imposed mathematical structure is called a forgetful functor. The following are examples of forgetful functors:

1. $U:\mathbf{Grp} \to \mathbf{Set}$ takes groups into their underlying sets and group homomorphisms to set maps.
2. $U:\mathbf{Top} \to \mathbf{Set}$ takes topological spaces into their underlying sets and continuous maps to set maps.
3. $U:\mathbf{Ab} \to \mathbf{Grp}$ takes abelian groups to groups and acts as identity on arrows.

Forgetful functors are often instrumental in studying adjoint functors.