Many forgetful functors in an algebraic setting are representable; for example, the forgetful functor $\text{Ab} \to \text{Grp}$ is represented by $(\mathbb{Z}, 1)$. Functors with left adjoints are also representable, and sometimes the converse holds.

Many forgetful functors in an algebraic setting are representable; for example, the forgetful functor $\text{Ab} \to \text{Set}$ is represented by $(\mathbb{Z}, 1)$. Functors with left adjoints are also representable, and sometimes the converse holds.