Forgetful functor (category theory)

Some categories have a forgetful functor $U:𝐂\to 𝐒𝐞𝐭$ which maps a category to its underlying set. One such category is $𝐆𝐫𝐩$, whose forgetful functor "forgets" the group structure leaving only the underlying sets of objects and plain functions in place of group homomorphisms.

There are "partially-forgetful" functors that forget some, but not all structure. Consider for example, an inclusion functor $\text{catname}Field\hookrightarrow \text{catname}Ring$, which forgets the structure of division, but preserves that of addition and multiplication.