The notion of "Cauchy real" is always a bit ambiguous: it depends on what you call a Cauchy sequence. For the argument that follow I need a notion of Cauchy sequence that is geometric (is classified by a locale). Two such examples (defining two different spaces of Cauchy Real) are:
A Cauchy sequence is a sequence of rational that converge (with just existential quantifier) in the space of Dedekind real.
A Cauchy sequence is a sequence of rational such that $|x_n - x_{n+1} |<1/n^2$ for all $n$.
The point is that with such a definition, the theory of Cauchy sequences is classified by a locale $K$ which is endowed with a map "limit" to the locale $\mathbb{R}$.
I claim that the map from $K$ to $\mathbb{R}$ is a really nice surjection. It might not be an open surjection by itself (and it is clearly not proper), but there is an open surjection $X \rightarrow \mathbb{R}$ that factors into $K$ (see below), hence $K \rightarrow \mathbb{R}$ is an effective descent morphism and in particular a stably regular epimorphism of locales.
Let $\mathcal{L}$ be a class of locales which contains both $K$ and $\mathbb{R}$ and is stable under open subspace and fiber product. And put a Grothendieck topology on it which is subcanonical and contain our surjection $K \rightarrow \mathbb{R}$ as well as the ope covering (the topology generated by open surjection, the canonical topology of $\mathcal{L}$, the canonical topology of the category of locale or the topology of effective descent morphism of locales are all acceptable solutions). let $\mathcal{T}$ be the resulting Grothendieck topos of sheaves.
It might not be true that the object of Dedekind real is represented by the locale $\mathbb{R}$. But if $X$ is a pro-discrete locale in $\mathcal{L}$ then the object of points of $X$ in $\mathcal{T}$ is represented by $X$ : Indeed this is clearly true if $X$ is a discrete locale and it pass to projective limit. Also it appears (see $C)$ below) that this is also true when $X$ is just a sub-locale of a pro-discrete locale. The locale for Cauchy sequences given above is pro-discrete so this apply to it, so the object of Cauchy sequences in $\mathcal{T}$ is represented by $K \in \mathcal{L}$.
The space of Cauchy real is in any topos described as the quotient of the object of points of $K$ by the equivalence relation of points of $K \times_{\mathbb{R}} K $ which is also a sublocale of a pro-discrete locale. The the object of Cauchy real is the quotient in $\mathcal{T}$ of the object represented by $K$ by the equivalence relation represented by $K \times_{\mathbb{R}} K $ but as we have chose the topology so that $K \rightarrow \mathbb{R}$ is a covering this quotient is the object represented by $\mathbb{R}$.
The object of endomorphisms of the object of Cauchy reals is then just the sheaf which to any locale $A \in \mathcal{L}$ associate the set of continuous functions from $A \times \mathbb{R} \rightarrow \mathbb{R}$ and they are internally continuous (to be honest, I havn't checked that last claim yet, but that sound reasonable).
Appendix: here are some clarification.
A) The open surjection $X \rightarrow K \rightarrow \mathbb{R}$.
We work internally in $\mathbb{R}$. One has the universal Dedekind real number $x$ corresponding to the identity $\mathbb{R} \rightarrow \mathbb{R}$.
For each positive integer $n$ the set $A_n$ of rational numbers $q$ such that $|x-q|<2^{-n}$ is inhabited (this is one of the characterization Dedekind numbers). you can then define the locale $X$ (still internally in $\mathbb{R}$) as $X =\prod_{n \in \mathbb{N}} A_n $. An inhabited set is in particular an open surjection to the point and it is a classical lemma in constructive locale theory that a product indexed by a decidable set of locales with an open surjection to the point also have an open surjection to the point (this is a kind of dual version of the Tychonov theorem for locales).
So we have constructed a locale $X$ with an open surjection to $\mathbb{R}$. You can check that the definition of the $A_n$ is geometric and hence stable under inverse image functor and that product of locales are stable under inverse image functors (obviously because inverse image functor are pullback in the category of locales). Hence, (externally now) $X$ classifies the theory whose model is the data of a real number $r$ together with a sequence of rational numbers $a_n$ such that for all $n$ $|a_n - r| < 2^{-n}$, the map to $\mathbb{R}$ just forgetting the sequence. The sequence of $a_n$ will then give you a Cauchy sequence converging to $r$ and hence you have a map from this locale to the locale $K$. (in fact depending on your definition of $K$ you might be able to prove this way that $K \rightarrow X$ itself is an open surjection.
B) The precise definiton of the locale $K$ for the two examples of definition of Cauchy sequences I gave above:
For the first one $K$ is the fiber product for $[\mathbb{N},\mathbb{Q}]$ and $[\mathbb{N}^{\infty},\mathbb{R}]$ over $[\mathbb{N},\mathbb{R}]$, where $\mathbb{Q}$ has the discrete topology and $\mathbb{N}^{\infty}$ is still the compact space with a sequence of points and its limit.
For the second definition, $K$ is just a (closed) subspace of the exponential locale $[\mathbb{N},\mathbb{Q}]$ defined by the inequalities in the definition.
C) For any sub-locale $X$ of a pro-discrete locale in $\mathcal{L}$ The object of $\mathcal{T}$-point of $X$ is the sheaves represented by $X$.
Let $U$ be an open subspace of $X$ a pro-discrete locale. Then it is still true that the space of points of $U$ in $\mathcal{T}$ is represented by $U$: indeed if $U$ is a basic open then $U$ is itself a pro-discrete locale, and otherwise $U$ is a union of basic open and this properties also passes to union as soon as we assume that the topology on $\mathcal{L}$ contains open covering.
Now because a pro-discrete locale is regular, any sublocale is an intersection of open sublocale and so can be written as a projective limit of locales which satisfies this properties.
D) What happen to Dedekind reals in $\mathcal{T}$ ?
SO this topos $\mathcal{T}$ as an object $\Omega'$ classified by the Sierpinski locale (whether it is in $\mathcal{L}$ or not) which is a subobject of the sub-object classifier. Because of that one dispose internally of a notion of "nice subobject" or "open subobject": by definition the sub-objects whose classifying map land in $\Omega'$, and hence internally $\Omega'$ si the object of nice subobject of $\{*\}$. One can easily see that $\mathbb{R}$ is exactly the object of "nice Dedekind real",that is of two sided dedekind cut, such that the two cut are nice subobject of $\mathbb{Q}$. But I don't see any reason for any two sidded dedekind cut to be "nice". In fact when we enlarge $\mathcal{L}$ the subobject classifier become larger and larger while $\Omega'$ stay roughly the same, hence one can image that the situation become worst and worst... but I havn't been able to find an example of a non nice Dedekind real, moreover the fact that the exact same discussion apply when the topology is just the topology of open covering and that in this case there is no "non nice" dedekind real suggest that one can still hope that the Dedekind real and the Cauchy real are the same.
