Let be the category whose objects are all commutative K-algebras with identity and whose morphisms are

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.