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

I think part of the confusion is that the sentence below the definition - "In the above diagram, $G := \Psi \circ F \circ \phi^{-1}: \phi(U \cap F^{-1}(V)) \rightarrow \Psi(V)$" - is not meant to be part of the definition, rather it is a clarification. The definition given is that $F$ is smooth iff there exists $G$ (which, from its position in the diagram, is a morphism between open subsets of $\mathbb{R^s}$, that is, is a smooth map*) such that
$\begin{CD}
M @> F > > N\\
@V \varphi V V @V  V\psi V\\
\varphi(U) @> G > > \psi(V)
\end{CD}$
commutes.
*- but I agree this should have been stated explicitly
Does that make things clearer at all?
