Let be the category of Exercise 1. If X is the set {x 1 ,

Let ℓ  be the category of Exercise 1. If X is the set {x₁, ••• , x_n}, then the polynomial algebra K[x₁, ... , x_n] is a free object on the set X in the category ℓ 

Data from exercise 1

Let ℓ  be the category whose objects are all commutative K-algebras with identity and whose morphisms are all K-algebra homomorphisms ∫: A → B such that ∫(1_A) = 1_B. Then any two K-algebras A, B of ℓ  have a coproduct.