Proof of direct sum of ideal class group of Neukirch book

Question

In books Neukirch, Algebraic Number Theory.

I don't understand.

1) Why there exists $a$ such that $a\equiv c \ \mod \mathfrak p $ and $a\in ca_{\mathfrak p}^{-1}a_{\mathfrak q}$ for $\mathfrak q\neq \mathfrak p$?

2) Why $\epsilon=ac^{-1}$ is a unit in $\mathcal O_{\mathfrak p}$?

Please.

Thanks you all-.

Answer 1

I asked exactly the same question two years ago: see here.
Also see my comment to the answer, which explains $(2)$.
Hope this helps.