structural:reformatting, stylistic:tone
Definition

Free objects are the direct generalization to categories of the notion of basis in a vector space. A linear function u : E → E between vector spaces is entirely determined by its values on a basis of the vector space E. The following definition translates this to any category.

A concrete category is a category that is equipped with a faithful functor to Set, the category of sets. Let C be a concrete category with a faithful functor U : C → Set. Let X be a set (that is, an object in Set), which will be the basis of the free object to be defined. A free object on X is a pair consisting of an object A {\displaystyle A} in C and an injection i: X → U (A) {\displaystyle i:X\to U(A)} (called the canonical injection), that satisfies the following universal property:

If free objects exist in C, the universal property implies every map between two sets induces a unique morphism between the free objects built on them, and this defines a functor F: S e t → C {\displaystyle F:\mathbf {Set} \to \mathbf {C} }. It follows that, if free objects exist in C, the functor F, called the free functor is a left adjoint to the faithful functor U; that is, there is a bijection