I have a problem with understanding the construction which I marked as the red ractangle. The whole concept is showing the alternative definition of a Thom space. Construction up to the second diagram is more or less clear, some of doubts I put into another question. Here I stuck on the last diagram. What role plays the "star" (*)? Also the sentence with 'zig-zag of weak equivalences' is completly incomprehensible to me.
Thanks in advance!