In the topos we construct in our paper there is a surjection/epimorphism from the natural numbers to the Dedekind reals. In the model of CZF you mention (and in the effective topos) the Dedekind reals are subcountable, meaning that there is a surjection from *a subset* of the natural numbers to the Dedekind reals. Externally this subset has non-trivial computational complexity (corresponding roughly to the set of Turing indices of total computable functions), so the situation in the new topos is more special.

One way to see how this is subtle is that it is known by [a result of Blechschmidt and Hutzler](https://arxiv.org/abs/1902.07366v1) that the MacNeille reals are uncountable in any topos. Also Cantor's diagonal argument and similar give that Cantor space and Baire space (and therefore the set of *irrational* Dedekind reals) are always uncoutnable, even when they are also subcountable. (And to clarify, I mean uncountable in the positive, constructive sense that given any sequence of elements one can construct an element not on the list.)

Incidentally given that the construction is a modification of the standard construction of the effective topos, it's a fairly trivial matter to modify it to give a construction of a model of IZF (and therefore also of CZF) in which the Dedekind reals are countable.