 Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.By definition, an adjunction between categories "C" and "D" is a pair of functors (assumed to be covariant)

and, for all objects "X" in "C" and "Y" in "D" a bijection between the respective morphism sets

such that this family of bijections is natural in "X" and "Y".The functor "F" is called a left adjoint functor or left adjoint to "G" , while "G" is called a right adjoint functor or right adjoint to "F" .