I am currently reading Zvonkine's "An Introduction to Moduli Spaces of Curves and Their Intersection Theory" and I am hoping that someone here would be willing to clarify some aspects of his definition of an orbifold subchart.
So if we have a topological space $X$, an open set $V \subset X$, and an orbifold chart $\varphi : U/G \to V$ where $U \subset \mathbf{R}^n, \mathbf{C}^n$ is a contractible open subset and $G$ acts smoothly/biholomorphically on $U$, then Zvonkine defines an orbifold subchart as follows:
Let $\varphi': U'/G' \to V'$ be a chart such that $V' \subset V$, $G'$ is group along with a group homomorphism $\theta: G' \to G$, and a smooth or holomorphic embedding $\iota: U' \to U$ where the following three conditions are satisfied:
The embedding and the group morphism commute with the group actions.
The $G'$-stabilizer of an element of $U'$ is isomorphic to the $G$-stabilizer of its image in $U$.
The embedding commutes with the isomorphisms $\varphi$ and $\varphi'$.
(2) has a very obvious interpretation. My question concerns what is meant by (1) and (3).
I am interpreting the statement "the embedding commutes with the group action" in (1) to mean that for $g' \in G'$ and $u_1',u_2' \in U'$, if $u_1' = g'u_2'$, then $\iota(u_1') = \theta(g')\iota(u_2')$.
I am somewhat unsure of what "the group homomorphism commutes with the group action" means in this context, as well as "the embedding $\iota: U' \to U$ commutes with $\varphi$ and $\varphi'$.
Any clarifications of what has been mentioned above would be greatly appreciated.