product category

Definition

• For \(C\) and \(D\) two categories, the product category \(C \times D\) is the category whose
  • objects are ordered pairs \((c, d)\) with \(c \in \text{Obj}(C)\) and \(d \in \text{Obj}(D)\).
    • \(\text{Obj}(C \times D) = \text{Obj}(C) \times \text{Obj}(D)\)
  • morphisms are ordered pairs \(((c \rightarrow{f} c'), (d \rightarrow{g} d'))\)
    • \(\text{Mor}(C \times D) = \text{Mor}(C) \times \text{Mor}(D)\)
  • composition of morphisms is defined componentwise by composition in \(C\) and \(D\).