The following is a question I had while reading Kirill Mackenzie's first book (RIP) on groupoids. He says that if a groupoid morphism is base-injective and piecewise-injective, then it is injective. I have a probably stupid question:
Let $\phi:G\rightarrow G'$, $\phi_0:B\rightarrow B'$ be a groupoid morphism. Suppose $\phi(\eta)=\phi(\xi)$ for some $\xi,\eta\in G$. Then clearly $\alpha'(\phi(\eta))=\alpha'(\phi(\xi))$ and also $\beta'(\phi(\eta))=\beta'(\phi(\xi))$ and so $\phi_0(\alpha(\xi))=\phi_0(\alpha(\eta))$ and $\phi_0(\beta(\xi))=\phi_0(\beta(\eta))$.
In other words, clearly $\phi(\eta)$ and $\phi(\xi)$ are arrows between the same base points, let's say $\phi_0(x)$ and $\phi_0(y)$. But if $\phi$ is piecewise injective, then every restriction, $\phi_x^y:G_x^y\rightarrow G_{\phi_0(x)}^{\phi_0(y)}$ is injective. In particular, since $\phi(\eta)=\phi(\xi)$ in $G_{\phi_0(x)}^{\phi_0(y)}$ and $\phi_x^y$ is injective, $\eta=\xi$.
Why must $\phi$ be base-injective then?