I'm starting to understand what commutative diagrams are, but I'm not sure about their purpose, what is their intended use and what kind of problems are solvable with them. By "solvable with a commutative diagram" I mean some fancy graphical reasoning, redrawing etc.
For example given only the commutative diagram for the exterior derivative
$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} \Omega^k(N) & \ra{f^*} & \Omega^k(M) \\ \da{d} & & \da{d} \\ \Omega^{k+1}(N) & \ras{f^*} & \Omega^{k+1}(M) \\ \end{array} $
is it even possible to tell that $d$ is a derivative, that is it is linear and the appropriate Leibniz rule holds?
Another example is my own, I may have done it totally wrong.
Given two vector spaces $V$ and $W$ (possibly of the same dimension) with scalar products, $f$ being a morphism, is it possible to prove that the following diagram commutes:
$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} V & \ra{f} & W \\ \da{h} & & \da{h} \\ V & \ras{f} & W \\ \end{array} $
only for $h$ of a certain form, which I guess is
$h(v) = \lambda(v^2) v$
where $\lambda$ is some arbitrary function? Is it a valid commutative diagram?