Let $\textbf{CartSp}$ be the category of spaces of the from $\mathbb{R}^n$ with smooth maps between them. This is a site with respect to (differentially) good open covers, so consider the Grothendieck topos of sheaves of sets on this site.

Now $\textbf{CartSp}$ is not just a site but also the syntactic category of the Lawvere theory of smooth algebras, namely those $\mathbb{R}$-algebras $A$ for which every smooth map $\mathbb{R}^n \to \mathbb{R}^m$ lifts to a map $A^n \to A^m$ in a compatible way.

That is to say, a smooth algebra is a product preserving copresheaf on $\textbf{CartSp}$ and $\textbf{Sh}(\textbf{CartSp})$ is the classifying topos of smooth algebras.

Now let $X$ be an object of $\textbf{Sh}(\textbf{CartSp})$. Then there is a naturally defined sheaf of smooth algebras$$\mathcal{O}_X: \textbf{CartSp} \to \textbf{Sh}(\textbf{CartSp}) \to \textbf{Sh}(\textbf{CartSp})/X$$which is supposed to be regarded as the structure sheaf.

So the outlook of this is that the subcategory of concrete and locally representable sheaves on $\textbf{CartSp}$ is equivalent to the category of smooth manifolds.

The trouble I am having is figuring out how to think of these as smooth manifolds. It would seem that such a sheaf $X$ really does not have the structure of a smooth manifold since it does not come with a structure sheaf.

Evidently I should instead be thinking of the slice topos $\textbf{Sh}(\textbf{CartSp})/X$ it induces, since this comes with a structure sheaf as I described earlier.

But I can not seem to figure out how to view $\textbf{Sh}(\textbf{CartSp})/X$ as a "space" in its own right. What are its open sets?

The motivation for this question is that I am trying to learn étale machinery from an abstract nonsense perspective but for smooth manifolds instead of schemes.