$C^{\ast }-$crossed product of groupoid actions on categories

Suppose that $G$ is a groupoid acting on a small category $H$ in the sense of \cite[Definition 4]{NOT} and $H{\times }_{\alpha }G$ is the resulting semi-direct product category (as in \cite[Proposition 8]{NOT}). We show that there exists a subcategory $H_r\subseteq H$ satisfying some nice property called regularity'' such that $H_r \times_\alpha G = H\times_\alpha G$. Moreover, we show that there exists a so-called quasi action’’ (see Definition \ref{quasi}) $\beta$ of $G$ on $C^{\ast }(H_r)$ (where $C^{\ast }(H_r)$ is the semigroupoid $C^{\ast }$-algebra as defined in \cite{EXE}) such that $C^(H_r\times_\alpha G) = C^(H_r)\times_\beta G$(wherethecrossedproductfor$\beta$ is as defined in Definition \ref{cross}).