In the definition of a smooth (or $C^k$) manifold the charts $\varphi: U \to \mathbb R^n$ are assumed to have the property that for any two of them $\varphi \circ \psi^{-1}$ is smooth ($C^k$).
Does $\varphi \circ \psi^{-1}$ is smooth ($C^k$) imply that both $\varphi$ and $\psi$ are smooth ($C^k$) or is it weaker?
It wouldn't make sense to say that $\phi$ or $\psi$ is smooth, because $U$ might not even have enough structure to make such a statement meaningful. $U$ might not be a subset of $\mathbb R^m$, for example. Even if $U$ is a subset of $\mathbb R^m$, typically $U$ would not be an open subset of $\mathbb R^m$.
$\phi \circ \psi^{-1}$, on the other hand, maps an open subset of $\mathbb R^n$ into $\mathbb R^n$, so it is perfectly meaningful to say that $\phi \circ \psi^{-1}$ is smooth.