Tree Automata Operators

I am trying to understand the Projection operation (linear tree homomorphism) and Cylindrification operation (inverse tree homomorphism) from the book. Linear Tree homomorphism is defined as follows:

Inverse Homomorphism is defined here as follows: With these two concepts defined, I try to use it to understand the following operations:

Now, there arent any step by step examples of how to accomplish ith projection/ ith cylindrification in tree automata in the referenced book. I would be grateful if you could point me to other useful references for these operations. That said, there is one example for the first projection, which I am unable to parse: How does one come up with the new set of transitions rules, and which symbol is being projected?