Hatcher's Proof of Van Kampen's theorem

Question

I have spent the last 5 days stuck on this last lines of Hatcher's proof of the Van Kampen's theorem:

My issue is this: why is any of this needed? Isn't it already established (by the definition of our partition of $I \times I$) that each $f_i$ is mapped into the corresponding $A_{\alpha}$?