If $X$ is a sober topological space, the real numbers object in the topos $\mathrm{Sh}(X)$ is the sheaf of continuous real-valued functions on $X$. This is proven very explicitly in Theorem VI.8.2 of MacLane & Moerdijk *Sheaves in Geometry and Logic* by compiling out the definition of real numbers in Kripke-Joyal semantics. A more abstract proof is in D4.7.6 of *Sketches of an Elephant*: since $\mathrm{Sh}(\mathbb{R})$ is the classifying topos of the theory of a real number, maps $U \to R\_D$ in $\mathrm{Sh}(X)$ from an open subset $U\subseteq X$ to the real numbers object are equivalent to geometric morphisms $\mathrm{Sh}(X)/U \to \mathrm{Sh}(\mathbb{R})$, but $\mathrm{Sh}(X)/U \simeq \mathrm{Sh}(U)$ and so these are equivalent to continuous maps $U\to \mathbb{R}$.


Theorem VI.9.2 of MacLane & Moerdijk makes an analogous claim for $\mathrm{Sh}(\mathbf{T})$, where $\mathbf{T}$ is a full subcategory of topological spaces closed under finite limits and open subspaces. My question is about a glossed-over point in the proof: they construct maps back and forth between sections of $R\_D$ and the continuous $\mathbb{R}$-valued functions, but they don't say anything about why the maps are inverses. I believe it in the situation of $\mathrm{Sh}(X)$, but for $\mathrm{Sh}(\mathbf{T})$ it's not obvious to me, because a continuous map $Y\to \mathbb{R}$ only "knows" about the open subspaces of $Y$, whereas a map $Y \to R\_D$ in $\mathrm{Sh}(\mathbf{T})$ says something about all continous maps $Z\to Y$ in $\mathbf{T}$. The most I can see how to show is that the sheaf of continuous real-valued functions is a retract of $R\_D$.


In terms of the abstract proof from the Elephant, the question is this: for $Y\in \mathbf{T}$, geometric morphisms $\mathrm{Sh}(\mathbf{T})/Y \to \mathrm{Sh}(\mathbb{R})$ are equivalent to continuous real-valued functions on the locale corresponding to the frame of subterminals in $\mathrm{Sh}(\mathbf{T})/Y$, but this frame is not (as far as I can see) the same as the open-set frame of $Y$.



The real numbers object in Sh(Top)
Following a [suggestion](http://nforum.ncatlab.org/discussion/6289/when-is-the-internal-real-line-the-external-real-line/?Focus=50368#Comment_50368) of Thomas Holder, we can close the gap as follows:


1. For each object $Y$ in $\mathbf{T}$, there is a pseudonatural local geometric morphism $\mathbf{Sh}(\mathbf{T}\_{/ Y}) \to \mathbf{Sh} (Y)$.
2. $\mathbb{R}$ is Hausdorff, so for any Grothendieck topos $\mathcal{E}$, the category of geometric morphisms $\mathcal{E} \to \mathbf{Sh} (\mathbb{R})$ is essentially discrete.
3. Essentially by definition, any local geometric morphism is a right adjoint in the bicategory of toposes. Thus, we have a pseudonatural left adjoint
$$\mathbf{Geom} (\mathbf{Sh} (Y), \mathbf{Sh} (\mathbb{R})) \to \mathbf{Geom} (\mathbf{Sh} (\mathbf{T}\_{/ Y}), \mathbf{Sh} (\mathbb{R}))$$
but any adjunction between groupoids is an equivalence, so we deduce that $\mathbf{T}(-, \mathbb{R})$ is not merely a retract of $R\_D$ in $\mathbf{Sh} (\mathbf{T})$ but actually isomorphic to $R\_D$, at least when every object in $\mathbf{T}$ is sober.


