My question is whether the axiom of extensionality is required to show that the schema of collection follows from the schema of replacement in the usual ZermeloFraenkel environment with choice. In other words: Is the schema of collection a theorem schema in ZermeloFraenkel set theory with choice minus the axiom schema of extensionality?
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Collection is not provable in ZFC minus extensionality, a simple countermodel is described in http://mathoverflow.net/questions/54328 . (That the model cannot provably satisfy collection follows from Gödel’s theorem. For a specific instance of collection which fails, let $\bar\omega$ denote one of the many representations of $\omega$ in the model, and $\bar0\in\bar\omega$ the corresponding empty set: then the model satisfies “for every $n\in\bar\omega\smallsetminus\{\bar0\}$, there exists a function $f$ with domain $n$ such that $f(\bar0)=\bar\omega$, and $f(x)\in f(y)$ whenever $x\in y\in n$”, but there is no set collecting such functions for every $n\in\bar\omega\smallsetminus\{\bar0\}$.) 

