Exercise 11.10 on page 122 of Bott and Tu asks us to show that, if $s:M\to E$ is a section (of a sphere bundle, in context), then $s^* K = K s^*$, where $K$ is the homotopy operator for the Cech complex of $M$ or $E$ (with good covers $U, \pi^{-1} U$, respectively).
I see that $$\pi \circ s = Id$$ gives $$s^* \circ \pi^* = Id,$$ and $K$ commutes with $\pi^*$ so that $$s^* K \pi^* = s^* \pi^* K = K = K s^* \pi^*,$$ but $\pi^*$ is surely not surjective (right?), so how do I see the desired equality?