Editing Symmetric algebra (section)

===From tensor algebra===
It is possible to use the [[tensor algebra]] {{math|''T''(''V'')}} to describe the symmetric algebra {{math|''S''(''V'')}}. In fact, {{math|''S''(''V'')}} can be defined as the [[quotient associative algebra|quotient algebra]] of {{math|''T''(''V'')}} by the two-sided ideal generated by the [[commutator]]s <math>v\otimes w - w\otimes v.</math>

It is straightforward to verify that the resulting algebra satisfies the universal property stated in the introduction. Because of the universal property of the tensor algebra, a linear map {{mvar|f}} from {{mvar|V}} to a commutative algebra {{mvar|A}} extends to an algebra homomorphism <math>T(V)\rightarrow A</math>, which factors through {{mvar|S(V)}} because {{mvar|A}} is commutative. The extension of {{mvar|f}}
to an algebra homomorphism <math>S(V)\rightarrow A</math> is unique because {{mvar|V}} generates {{mvar|S(V)}} as a {{mvar|K}}-algebra.

This results also directly from a general result of [[category theory]], which asserts that the composition of two [[left adjoint]] functors is also a left adjoint functor. Here, the [[forgetful functor]] from commutative algebras to vector spaces or modules (forgetting the multiplication) is the composition of the forgetful functors from commutative algebras to associative algebras (forgetting commutativity), and from associative algebras to vectors or modules (forgetting the multiplication). As the tensor algebra and the quotient by commutators are left adjoint to these forgetful functors, their composition is left adjoint to the forgetful functor from commutative algebra to vectors or modules, and this proves the desired universal property.