Structomorphisms and diffeomorphisms are equivalent while preserving the underlying homeomorphism

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. There exists a bijection $\Phi$ from the type of homeomorphisms $h:M\to M'$ equipped with the condition that every atlas coordinate expression $c'\circ h\circ c^{-1}$ and its inverse become $C^n$ after conjugation by $I$ on their respective model-coordinate domains, to the type of bijections $f:M\to M'$ for which both $f$ and $f^{-1}$ are $C^n$ manifold maps relative to $I$, such that for every such $h$ the homeomorphism underlying $\Phi(h)$ is exactly the homeomorphism underlying $h$. Being a $C^n$ manifold map here means being continuous at every domain point $x$ and having the extended preferred-chart expression $\kappa_{f(x)}\circ f\circ\kappa_x^{-1}$ of class $C^n$ at $\kappa_x(x)$ within $I(H)$, where $\kappa_x=I\circ c_x$. The claimed bijection is an equivalence of the two types, with no further algebraic or topological structure imposed on that equivalence and no uniqueness assertion. Neither type of map is required to be inhabited, so the statement includes the case of two empty types of maps; it also includes empty underlying spaces, empty chart overlaps, and $n=0$, and assumes neither finite-dimensionality nor completeness of $E$.