Exercise 5, § 3

a) Let $\varphi : G\rightarrow G'$ be a morphism of groupoids such that the mapping Som($\varphi$ ) is injective. Prove that $\varphi (G)$ is a subgroupoid of $G'$.

b) Let G be a non-transitive groupoid whose set of vertices Som(G) is a set $\{a, b\}$ with two elements. Let $G'$ be the groupoid associated with the free product $G_a*G_b$. There exists a unique morphism of groupoids $\varphi : G\rightarrow G'$ such that the mappings $\varphi_a$ and $\varphi_b$ are the canonical mappings of $G_a$ and $G_b$ into $G_a*G_b$. If $G_a$ or $G_b$ is not the trivial group, the image $\varphi (G)$ of this morphism of groupoids is not a subgroupoid of $G'$.