Let $(X,\mathcal{O}_X)$ be a (pre)manifold, defined as a ringed space locally isomorphic to $(\mathbb{R}^n,\mathcal{O})$. Here, $\mathcal{O}=C^{\infty}$ as the usual sheaf.
In particular, there is a covering $\mathcal{U}=\lbrace\mathcal{U}_i\rbrace$ of $X$ such that $(\mathcal{U}_i,\mathcal{O}_{X}(\mathcal{U_i}))$ is isomorphic (as ringed space) to some $(\mathcal{V}_i,\mathcal{O}(\mathcal{V_i}))$, where $\mathcal{V}_i$ is an open subset of $\mathbb{R}^n$.
This question is very simple, yet I am getting confused: If we take as coordinates $x_i:\mathcal{U}_i\rightarrow\mathcal{V}_i$, where $x_i$ is a homeomorphism, how can I guarantee that the transition maps will be $C^{\infty}$ ? That way, I would have the more classical structure of coordinates from this definition.