Definition 43.1.8 (Pointed Flow).label Let $G$ be a group, then a pointed $G$-flow is a pair $(X, x_{0})$, where $X$ is a $G$-flow, and $x_{0} \in X$ is a distinguished point. The mapping $\rho_{x}: G \to X$ defined by $g \mapsto gx_{0}$ is the orbit map of $(X, x_{0})$.

Let $(G \curvearrowleft X, x_{0})$ and $(G \curvearrowright Y, y_{0})$ be pointed $G$-flows, then a morphism of pointed $G$-flows is a morphism of $G$-flows $\pi: X \to Y$ such that $\pi(x_{0}) = y_{0}$.