Product category

Summary

The product category $\mathcal{C} \times \mathcal{D}$ of two categories has pairs of objects as objects and pairs of morphisms as morphisms, composed in each coordinate. It is the product of $\mathcal{C}$ and $\mathcal{D}$ in the category of small categories. It is needed to speak of functors of two variables, such as the hom functor $\Hom(-, -)$ on $\mathcal{C}^{\mathrm{op}} \times \mathcal{C}$ or the tensor product of modules.