morphism from to is given by a morphism in such that the diagram commutes: Now suppose that the object in is initial. Then for every object , there exists a unique morphism such that the following diagram commutes. Note that the equality here simply means the diagrams are the same. Also note that the diagram on the right side of the equality is the exact same as the one offered in defining a universal morphism from to . Therefore, we see that a universal morphism from to is equivalent to an initial object in the comma category . Conversely, recall that the comma category is the category where Objects are pairs of the form where is an object in A morphism from to is given by a morphism in such that the diagram commutes: Suppose is a terminal object in . Then for every object , there exists a unique morphism such that the following diagrams commute. The diagram on the right side of the equality is the same diagram pictured when defining a universal morphism from to . Hence, a universal morphism from to corresponds with a terminal object in the comma category . Examples Below are a few examples, to highlight the general idea. The reader can construct numerous other examples by consulting the articles mentioned in the introduction. Tensor algebras Let be the category of vector spaces -Vect over a field and let be the category of algebras -Alg over (assumed to be unital and associative). Let : -Alg → -Vect be the forgetful functor which assigns to each algebra its underlying vector space. Given any vector space over we can construct the tensor algebra . The tensor algebra is characterized by the fact: “Any linear map from to an algebra can be uniquely extended to an algebra homomorphism from to .” This statement is an initial property of the tensor algebra since it expresses the fact that the pair , where is the inclusion map, is a universal morphism from the vector space to the functor . Since this construction works for any vector space , we conclude that is a functor from -Vect to -Alg. This means that is left adjoint to the forgetful functor (see the section below on relation to adjoint functors). Products A categorical product can be characterized by a universal construction. For concreteness, one may consider the Cartesian product in Set,