  This is the second in a series of papers on the relation between algebraic
set theory and predicative formal systems. In part I, we introduced the notion
of a predicative category of small maps and obtained the result that such
categories always contain a model of set theory. In the present paper, we show
that the familiar realizability models of the constructive set theories CZF and
IZF can be obtained as an application of this result. For this purpose, we show
that predicative categories with small maps are closed under an internal notion
of realizability.
