I am trying to prove the corollary of the Covering Homotopy theorem
Let $B_0$ be paracompact. Let $F:\xi_0 \rightarrow \xi_1$ be a morphism of vector bundles covering $f:B_0\rightarrow B_1$. Let $h:B_0\times I\rightarrow B_1\times I$ be a homotopy of $f$.Then there is a morphism $H:\xi_0\times I\rightarrow \xi_1\times I$ of $F$ covering $h$.
So I am having some trouble proving this , I know it's enough to find a morphism $\xi_0\times I\rightarrow \xi_1\times I$ wich covers $g:B_0\times I\rightarrow B_1\times I$ such that $g(x,t)=(h(x,t),t)$. I also tried using the covering homotopy theorem in the proof but I couldn't get anything useful , so any help with this is aprecciated. Thanks in advance!