I am trying to solve the problem 180 on Davis-Kirk (p.269). However, the authors give some special definitions, so-called "stable system of vector bundles", which I have never seen on other literature. So I interpret the question in the following way.
(If you are interested, the definition is on page 263.)
Given a collection of vector bundles $\gamma_k:E_{k}\to B_k$ of rank $k$ such that we have the collection of bundle maps $g_k$ and $\tilde{g_k}$ such that the diagram commutes:
$\hskip2in$ 
Now, we have such a collection and the classifying maps $f_k:B_k\to BO(k)$ of $\gamma_k$. Then we have the diagram below, where the left face is commutative; the right, back, and front face are pull-back diagrams.
We want to show that the bottom and top face commute.
$\hskip2in$ 
I tried to use the universal property of pull backs and different ways of "diagram chasing", none of them work.