information:classification, information:filtering
Existence

Not every functor G : C → D admits a left adjoint. If C is a complete category, then the functors with left adjoints can be characterized by the adjoint functor theorem of Peter J. Freyd: G has a left adjoint if and only if it is continuous and a certain smallness condition is satisfied: for every object Y of D there exists a family of morphisms

where the indices i come from a set I, not a proper class, such that every morphism

can be written as

for some i in I and some morphism

An analogous statement characterizes those functors with a right adjoint.

An important special case is that of locally presentable categories. If F: C → D {\displaystyle F:C\to D} is a functor between locally presentable categories, then