Definition 4.2.20. Let $\mathcal{A}$, $\mathcal{B}$ be categories. We define the *product category* $\mathcal{A} \times \mathcal{B}$ to be the category with objects $\mathop{\mathrm{Ob}}\nolimits (\mathcal{A} \times \mathcal{B}) = \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}) \times \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ and

Composition is defined componentwise.

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