An atlas can be defined as a set of $\{U_\alpha,\phi_\alpha\}$, where $U_\alpha \subseteq S$ are subsets of a topologically suitable set $S$ and $\phi_\alpha : U_\alpha \rightarrow V_\alpha \subseteq\mathbb R^n$ are invertible functions that give coordinates to the subsets. Sometimes, it can be useful to take an "internal" view, where we forget that the points in $S$ belong to a larger space in order to focus on what things look like "inside" the manifold. In practice, I have seen "inside view atlases" given in forms like, $(\{V_\alpha\ \subseteq R^n\},\{\varphi_{\alpha\beta} : U_\alpha \subseteq V_\alpha \rightarrow U_\beta \subseteq V_\beta\})$: a set of subsets of Euclidian space, along with the connecting functions $\varphi_{\alpha\beta}$ that link them all together. It's easy to go from the first view to the second view, because it is clear that $\varphi_{\alpha\beta} = \phi_\beta \circ \phi_\alpha^{-1}$ will work. However, after thinking about it, I have realized that it is not so clear that you can always go in the other direction.
In the second way of writing atlases, it is possible to have contradictions. The way I have written it, it would be permitted that $\varphi_{12}\varphi_{21}x\neq x$, although it is easy to prevent this by adding a condition that $\varphi_{\alpha\beta}^{-1} = \varphi_{\beta\alpha}$. Still, the problems have not all gone away. It is possible for $\varphi_{12}\varphi_{23}\varphi_{31}x \neq x$, which would not be forbidden by the inverse requirement. It appears that thinking up specific rules to block off particular counterexamples will not be sufficient.
So, I would like to know what conditions I have to apply in the "inside view," if I want to guarantee that an atlas specified in that way is really a manifold. How can I do this?