Checking that a $3$-D diagram is commutative

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?

The most common way to get out of checking all the small squares is to have some monic or epic arrows in your diagram. For instance, suppose we knew the square from $B_1$ to $C_2$ was commutative, that $B_2\to C_2$ was monic, and that the large rectangle from $A_1$ to $C_2$ was commutative (that's a bit strong here since it's already epic, but this is discussion applies more generally.) Then by canceling the monic arrows we could deduce commutativity of the square from $A_1$ to $B_2$. Of course, you can dualize, and you can use this on cubes as well as rectangles: if you knew the top, bottom, front, back, and right faces of the $A,B$ cube were commutative then $A_2'\to B_2'$ being monic implies the left face is commutative.

But in general, even checking all but one square does not suffice, as you can see in your diagram: consider setting the $A_i'$ and $B_i'$ and all four $C$s to zero (then everything but the top left square automatically commutes, but we know nothing about the latter.)

If you are writing the proof, then you should at least explain how to check each one (maybe give a single example). If all the remaining are similar, then you can just say they are similar.

The really important thing is that you personally verified each fact. If you did not do it personally, how can you assert that you know it is commutative? If you can prove to yourself that it is true without checking each individual one, then you should be able to write the proof without checking each individual one.