Corollary 8.8.5

[blindFS]

"the hypotheses imply that $\pi_k(fib_f {f(a)})=0$ for all $k\le n$ and $a:A$, and that $||{fib_f{f(a)}}||_0$ is contractible. Since $\pi_k(fib_f {f(a)}) = \pi_k(||{fib_f{f(a)}}||_n)$ for $k\le n$, and $||{fib_f{f(a)}}||_n$ is $n$-connected, by 8.8.4 it is contractible for any $a$."

I have following questions about this proof:

1. If $||{fib_f{f(a)}}||_n$ is $n$-connected, then it is contractible by definition of n-connectedness, why do we need 8.8.4? My understanding is that here the "n-connected" should be "a n-type", matching the presumption of corollary 8.8.4
2. How to prove the following statement $\pi_k(fib_f {f(a)}) = \pi_k(||{fib_f{f(a)}}||_n)$ for $k\le n$? The closest lemma I can find in this book is lemma 8.3.2, which again requires that $||{fib_f{f(a)}}||_n$ is $n$-connected, i.e. the missing part of the proof
3. Where is the assumption (iii) used in this proof?