product category

Definition

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