What does it mean to say a diagram commutes? $\require{AMScd}$
In the context of smooth manifolds, the map $F:M\rightarrow N$ is smooth if $G$ on the below diagram is smooth.
$\begin{CD}
M @> F > > N\\
@V \varphi V V @V  V\psi V\\
\varphi(U) @> G >=\psi\circ F\circ\varphi^{-1} > \psi(V)
\end{CD}$
Some notes I found say "if and only if this diagram commutes" however this diagram always commutes! if $F$ isn't smooth all it means is $G$ wont be smooth.
What does it mean to say a diagram commutes in this case?
Notes:
http://www.math.toronto.edu/mat1300/smoothmaps.4.pdf
Definition:
http://www.maths.kisogo.com/index.php?title=Smooth_map

"The diagram commutes" means exactly what it always means: that the map produced by following any path through the diagram is the same.
The notes you are reading made a mistake and forgot to require that $G$ be smooth.
