Let $\mathcal{G}\rightrightarrows M$ be a groupoid and $G$ be a group acting on $M$. Then we might associate the action groupoid $G\ltimes M\rightrightarrows M$.
What are the groupoid morphisms from $\mathcal{G}\rightrightarrows M$ to the action groupoid?
I got to the conclusion that if $(F, f)$ is such a morphism then $F=(F^\prime, F^{\prime\prime})$ where $F^\prime:\mathcal{G}\longrightarrow G$ satisfies $$F^{\prime}(gh)=F^{\prime}(g)F^\prime(h)\quad \textrm{and}\quad F^\prime(1_x)=e_G,$$ and $F^{\prime\prime}:\mathcal{G}\longrightarrow M$ satisfies: $$F^{\prime\prime}(gh)=F^{\prime\prime}(g)\quad\textrm{and}\quad F^{\prime\prime}(1_x)=f(x).$$ Therefore $(F^\prime, f^\prime)$ is a morphism from $\mathcal{G}\rightrightarrows M$ to $G\rightrightarrows \{*\}$ where $f^\prime:M\longrightarrow\{*\}$ is the constant map $x\longmapsto *$.
What about $F^{\prime\prime}$, is it also a morphism between suitable groupoids?