Structomorphisms are differentiable

Let $\mathbb k$ be any nontrivially normed field, let $E$ be a normed additive commutative group with a normed $\mathbb k$-vector-space structure, let $H$ be a topological space, and let $I$ be a model with corners from $H$ to $E$: its forward map is a continuous embedding defined on all of $H$, its inverse is a continuous total map $E\to H$ inverse on the range, and its range has nonempty interior; that range is convex over the induced real scalars when $\mathbb k$ has a compatible real- or complex-field structure, and is all of $E$ otherwise. Let $n$ be any nonnegative integer, $\infty$, or $\omega$, denoting respectively finite differentiability order, smoothness, or analyticity, and let $M,M'$ be topological spaces with chosen atlases and preferred local charts into $H$, both satisfying the $C^n$ manifold compatibility condition for $I$. Here that compatibility means every atlas transition $\psi$ and its inverse become $C^n$ in $E$ after conjugation by $I$, on $I(\operatorname{source}\psi)$ and $I(\operatorname{target}\psi)$ respectively. For every homeomorphism $h:M\to M'$ such that, for every atlas chart $c$ of $M$ and $c'$ of $M'$, the partial coordinate map $\psi=c'\circ h\circ c^{-1}$ and its inverse satisfy those same $C^n$ conditions after conjugation by $I$ on their respective domains, the underlying map $h$ is $C^n$ as a manifold map relative to $I$ on both sides. Explicitly, at each $x\in M$ it is continuous at $x$, and, writing $\kappa_x=I\circ c_x$ for the extended preferred chart at $x$, the coordinate expression $\kappa_{h(x)}\circ h\circ\kappa_x^{-1}$ is $C^n$ at $\kappa_x(x)$ within $I(H)$. The order $n=0$ is included; there is no finite-dimensionality or completeness assumption on $E$. Empty coordinate overlaps impose vacuous conditions, and an empty $M$ or $M'$ is allowed, with the assumed homeomorphism then forcing both to be empty.