When proving certain results I need to use commutative diagrams, some of which quite complicated. My question is:
Do we need to check every small square all the time to make sure that they are all commutative?
As an example, if we have the following diagram. If in my proof I wrote "Consider the following commutative diagram":
Before discussing anything else, I need to prove that it is indeed commutative. There are $11$ small squares to verify. When reading papers/books, I seldom see the author verifies every small square is commutative.
Is there any alternative other than checking all small squares, if I want to claim that a complicated diagram is commutative?