TITLE: Geometric intuition for Mather's cube theorem
QUESTION [5 upvotes]: Mather's cube theorem for the category of topological spaces says that given a homotopy-commutative cube:

If one pair of opposite faces are homotopy pushouts and the two
  remaining faces adjecent the source vertex are homotopy pullbacks,
  then the final two faces are also homotopy pullbacks.

What is the geometric intuition behind this theorem?

REPLY [4 votes]: I'm not sure what counts as an intuitive explanation, but this is sort of how I think about it.
Say that $B=B_1\cup B_2$ and $B_0=B_1\cap B_2$. This is the second pushout square.
Now let $E_1$ be a bundle over $B_1$ and let $E_2$ be a bundle over $B_2$, and suppose that the restriction to $B_0$ is the same for both bundles -- call it $E_0$. These are the two given pullback squares. Now let $E$ be the union of $E_1$ and $E_2$ along $E_0$. This is the other pushout square. $E$ should be a bundle over $B$ whose restriction to $B_1$ (resp. $B_2$) is the bundle $E_1$ (resp. $E_2$).