because X is a set, while F(X) is an algebra; correctly, it is ). The natural transformation is called the unit; together with the counit , one may construct a T-algebra, and so a monad. The cofree functor is the right adjoint to the forgetful functor. Existence There are general existence theorems that apply; the most basic of them guarantees that Whenever C is a variety, then for every set X there is a free object F(X) in C. Here, a variety is a synonym for a finitary algebraic category, thus implying that the set of relations are finitary, and algebraic because it is monadic over Set. General case Other types of forgetfulness also give rise to objects quite like free objects, in that they are left adjoint to a forgetful functor, not necessarily to sets. For example, the tensor algebra construction on a vector space is the left adjoint to the functor on associative algebras that ignores the algebra structure. It is therefore often also called a free algebra. Likewise the symmetric algebra and exterior algebra are free symmetric and anti-symmetric algebras on a vector space. List of free objects Specific kinds of free objects include: free algebra free associative algebra free commutative algebra free category free strict monoidal category free group free abelian group free partially commutative group free Kleene algebra free lattice free Boolean algebra free distributive lattice free Heyting algebra free modular lattice free Lie algebra free magma free module, and in particular, vector space free monoid free commutative monoid free partially commutative monoid free ring free semigroup free semiring free commutative semiring free theory term algebra discrete space See also Generating set Notes Mathematics articles needing expert attention Abstract algebra Combinatorics on words Adjoint functors