I am currently studying Differential Topology and I've noticed that commutative diagrams keep on popping up over and over.
An example is the following snippet from Topology from the Differential Viewpoint by Milnor :
All I know about commutative diagrams at the moment is that that they originate in Category Theory, however looking up introductory texts on Category Theory do not give a good definition of Commutative Diagrams, and hence I have no idea how they are used.
Where can I learn the necessary Category Theory on commutative diagrams to be able to understand such arguments used in Differential Topology?
A commutative diagram is nothing but a way to state briefly and visually that certain compositions of functions are equal. There is absolutely nothing more to them than that.