Exercise 1, § 3

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

(a) Now prove that the quiver C, endowed with the composition law deduced from the composition laws of the categories $C_i$, is a category. It is said that this is the sum category of the family $(C_i)$. (b) Prove that the canonical morphism from $C_i$ into C is a functor. (c) If the $C_i$ are groupoids, prove that the category C is a groupoid.