Exercise 2, § 3

Let $(C_i)_{i\in I}$ be a family of categories. Let C be the product of the underlying quivers of the categories $C_i$.

a) Now prove that the quiver C, endowed with the law of composition deduced from the laws of composition of the categories $C_i$, is a category. We say that it is the product category of the family $(C_i)$.

b) Exercise prove that the canonical morphism pr$_i: C\rightarrow C_i$ is a functor.

c) If the $C_i$ are groupoids, exercise prove that the category C is a groupoid.