Why is the pasting of commutative diagrams commutative?

Question

Wikipedia says the following about commutative diagrams:

Commutativity makes sense for a polygon of any finite number of sides (including just 1 or 2), and a diagram is commutative if every polygonal subdiagram is commutative.

However it is not obvious to me that a diagram is commutative if every polygonal sub diagram is commutative. For example consider two commutative squares pasted together with arrows going to the bottom right of each square. I can't find a way to show the commutativity of the big square using the commutativity of the smaller ones.

Answer 1

Start with the path that goes right-right-down. Applying the commutativity of the right-most square gives you a new path through the diagram: right-down-right. Now taking that path and applying commutativity of the left-most square gives down-right-right. These are the three paths through the diagram, so the diagram commutes.