Models of intuitionistic set theory in subtoposes of nested realizability toposes

In [8] Joyal and Moerdijk have shown that realizability toposes over partial combinatory algebras (pca) host classes of small maps   giving rise to initial ZFZF-algebras providing models of intuitionistic Zermelo–Fraenkel set theory IZFIZF. Here we show that this can be extended to a much wider class of realizability toposes as considered in [4] and [3].For this purpose we first show this result for nested realizability toposes  RT(A,A#)RT(A,A#) induced by a pca AA together with a sub-pca A#A# as considered implicitly in [4] and then show that it is preserved by restriction to subtoposes. This suffices since all toposes considered in [4] and [3] arise as subtoposes of some nested realizability toposes.