(This is a follow-up question to this earlier one.)
Setup: let $u : \mathrm{Diff} \rightarrow\mathrm{Set}$ denote the forgetful functor on the category of smooth manifolds.
Let $\tilde{X} \subset \mathrm{Hom}_{\mathrm{Set}}(u(-),X_{0})$ be a diffeology (per the nLab definition) on a set $X_{0}$, called the "underlying set" of $\tilde{X}$. Note $\tilde{X} \in \mathrm{Functors}(\mathrm{Diff}^{\mathrm{op}}\rightarrow \mathrm{Set})$.
Let $X = \mathrm{Dtopology}(\tilde{X}) \in \mathrm{DTopologicalSpaces}$ be the underlying D-topological space of $\tilde{X}$ (cf. the above nLab page).
Note $X$ is the topological space with underlying set $X_0$ of $\tilde{X}$, equipped with the final topology induced by all the maps $f\in\tilde{X}(M) \subset \mathrm{Hom}_{\mathrm{Set}}(uM,X_0)$ for all $M \in \mathrm{Diff}$.
According to nLab, the $X$ above is then a $\Delta$-generated topological space, or D-topological space or D-space for short.
Let us define a presheaf $\mathcal{F}_{\tilde{X}}$ on $X$ via $$ \mathcal{F}_{\tilde{X}}(U) = \left\{f : U \rightarrow \mathbb{R}\,\big|\, \forall M \in \mathrm{Diff} , \forall g \in \tilde{X}(M) \subset \mathrm{Hom}_{\mathrm{Set}}(M,X_0) ,\, \bigl(\,g^{-1}(U)\xrightarrow{f\circ g} \mathbb{R} \text{ is smooth} \,\bigr) \right\} $$ as a (unital) $\mathbb{R}$-algebra subpresheaf of the sheaf of continuous $\mathbb{R}$-valued functions on $X$.
The presheaf $\mathcal{F}$ is a sheaf and has local stalks, making $X$ a locally ringed space. Let us call $\mathcal{F}_{\tilde{X}}$ the "sheaf of diffeologically-smooth functions" on $X$.
Furthermore the above construction $\tilde{X} \mapsto (X,\mathcal{F}_{\tilde{X}})$ gives a functor $$ \psi : \mathrm{DiffeologicalSpaces} \rightarrow \mathrm{LocallyRingedDSpaces} $$
- Following an answer posted on the question linked at the top, we can see that $\psi$ is faithful. Indeed, the forgetful functor from RingedSpaces to Spaces, composed after $\psi$, equals the Dtopology functor $\mathrm{DiffeologicalSpaces} \xrightarrow{\tilde{X} \mapsto X} \mathrm{DSpaces}$, which the aforementioned answer shows is already faithful.
My question now is: does $\psi : \mathrm{DiffeologicalSpaces} \rightarrow \mathrm{LocallyRingedDSpaces}$ reflect isomorphisms? (Such functors are also called "conservative.")
I.e., if $\tilde{f} : \tilde{X} \rightarrow \tilde{Y}$ is a morphism of diffeological spaces, and $\psi(\tilde{f})$ is an isomorphism (in the category LocallyRingedSpaces), then is $\tilde{f}$ an isomorphism of diffeological spaces?
If I'm not mistaken, $\tilde{f}$ can be a non-isomorphism while $\mathrm{Dtopology}(\tilde{f}) = f : X\rightarrow Y$ is a homeomorphism. (So, the Dtopology functor does not reflect isomorphisms.)
(Edit: changed "detect" to "reflect" and added the nLab link.)