Some remarks on groupoids and small categories

This unpublished note contains some materials taken from my old study note on groupoids and small categories. It contains a proof for the fact that any groupoid is a group bundle over an equivalence relation. Moreover, the action of a category $G$ on a category $H$ as well as the resulting semi-direct product category $H{\times }_{\alpha }G$ will be defined (when either $G$ is a groupoid or $H^{(0)}=G^{(0)}$). If both $G$ and $H$ are groupoids, then $H{\times }_{\alpha }G$ is also a groupoid. The reason of producing this note is for people who want to check some details in a recent work of Li.