  This unpublished note contains some materials taken from my old study note on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.
