Question: I am having trouble understanding where the definition of $\bar{\omega}$ comes from. I understand that it should be a $2$-form on $cL_s$ undergoing pullback. With this said, why is the argument $\bar{\omega}([z,t])$ instead of just $\bar{\omega}(z,t)$? How does the rest of $(4.1)$ come into play?
Reference: The link to these notes is here.
Update: After some more careful reading, I found that $\bar{\omega}([z,t])$ is the $(0,2)$-tensor at the point $[z,t] \in cL_s(\epsilon_s)$, where $[z,t]$ is the image of $(z,t)$ under the natural projection $\pi: L \times [0, \infty) \to cL$. I still need help understanding the right hand side of $(4.1)$.