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 faithful functor . Let "X" be an object in Set (that is, "X" is a set, here called a "basis"), let "A" be an object in C, and let be an injective map between the sets "X" and "F"("A") (called the "canonical insertion"). Then "A" is said to be the free object on "X" (with respect to "i") if and only if it satisfies the following universal property: