Proving that a map between manifolds is smooth via a covering set of coordinate systems

Question

Warner's book defines a $C^\infty$ map between manifolds as follows: "A continuous map $\psi : M \to N$ is said to be $C^\infty$ if $\phi \circ \psi \circ \tau^{-1}$ is $C^\infty$ for each coordinate map $\tau$ on $M$ and $\phi$ on $N$." To prove a map to be $C^\infty$ using this definition, we obviously can't test the condition on all pairs of coordinate maps, because there are absurdly many. Is it sufficient, however, to choose a set of coordinate systems on $M$ and $N$ that cover $M$ and $N$, respectively, and show that the condition holds for pairs of these?