I've been reading through Greenblatt's Topoi and while I'm still definitely over my head I'm starting to get a feel for some of the concepts at play there. I see the definitions of $\mathbb{R}_c$ and $\mathbb{R}_d$ in a topos and they somewhat make sense to me, but there's another definition that seems 'natural' that I'm surprised not to find.

Specifically, it seems as though we could define the reals as the (Hom-)set of order-preserving maps $r:\mathbb{Q}\to\Omega$, or more narrowly the subobject here which consists of all such maps such that $r^{-1}(0)$ and $r^{-1}(1)$ are both inhabited. Given that I haven't seen anything like this I'm presuming that there are conceptual issues with it; I would guess that constructibility is a problem but again I'm very close to drowning in these waters so my knowledge is a bit weak here. Has this definition of reals been looked at at all, and what are the problems with it?