Exercise 3-90

  1. What are identity morphism in a product category \(\mathcal{C}\times \mathcal{D}\)?

  2. Why is composition in a product category associative?

  3. What is the product category \(1 \times 2\)?

  4. What is the product category of two preorders?

Solution(1)

  1. For object \((c,d)\), the identity morphism is \((id_c,id_d)\)

  2. The operation was defined in terms of function composition which is associative.

  3. It is isomorphic to just 2

  4. The underlying set is the cartesian product, and \((a,b)\leq(a',b')\) iff \(a \leq a' \land b \leq b'\)