I am read the Lemma 2 (pp13) in the paper "Kazukuni Kobara and Hideki Imai: Semantically Secure McEliece Public-Key Cryptosystems –Conversions for McEliece PKC– (PKC 2001)".
Related to the question "Why… for any $Hash_z$ and any $Gen$?", the author of the paper replies;
The reason why "for any $Hash_z$ and any $Gen$" is that if the sentence is not included the cryptosystem may be broken due to weak $Hash_z$ or $Gen$.
I know the hash strong definition:
There exist no x and x' with x != x' so that h(x) = h(x')
In this case are $x=Hash_z$, $x'=Gen$" and $h=A$?
Are the $Hash_z$ and $Gen$" from Lemma 2 the same that $Hash_z$ and $Gen$" of Figure 5 of that paper?