I am reading chapter 7 lemma 2.2 in "Differential topology" by M.Hirsch which is about Thom's isomorphism. The lemma goes as
Lemma 2.2: Let $Q$ be a compact manifold, and $B$ be a compact manifold without boundary. Then every map $f:Q\to E^*$ is homotopic to a map in standard form.
Here $E\to B$ is a vector bundle, and $E^*$ is the Thom space i.e. the one point compactification of $E$. A map $h:Q\to E^*$ is said to be in standard form if there is a submanifold $M\subset Q$ and a tubular neighborhood $U$ of $M$ such that $U=h^{-1}(E)$ and $M=h^{-1}(B)$ and $h|U$ is a vector bundle map i.e. $h|U$ is ismorphism on each fibre.
The proof of the lemma goes like:
By a preliminary homotopy, we may assume $f\pitchfork B$. Put $M=f^{-1}(B)$, choose $U\subset f^{-1}(E)$ a tubular neighborhood of $M$ and $D\subset U$ a disk subbundle. By theorem 4.6.5 we can assume that $f$ agrees in $D$ with a vector bundle map $\Phi:U\to E$. Define a map $h : Q \rightarrow E^{*}$ $h=\left\{\begin{array}{ccc}{\Phi} & {\text { on }} & {U} \\ {\infty} & {\text { on }} & {Q-U}\end{array}\right.$ Then $h$ agrees with $f$ on $D,$ and $h$ is in standard form. $\Phi : U \rightarrow E . $Since $h$ and $f$ agree on $\partial D,$ and both map $Q-$ int $D$ into contractible space $E^{*}-B,$ it follows from Lemma 2.1 that $f \simeq h .$
I think I get stuck on the bold sentence in which I don't know how theorem 4.6.5 can lead to the assumption that $f$ agrees with a vector bundle map or that $f$ is homotopic to a vector bundle map. Theorem 4.6.5 is:
Theorem. 6.5. Let $M \subset V$ be a submanifold. Let $\xi_{i}=\left(p_{i}, E_{i}, M\right)$ be orthogonal vector bundles over $M, i=0,1 .$Let$\left(f_{i}, \xi_{i}\right)$ be a tubular neighborhood of $M .$ Let $\varepsilon>0, \delta>0 .$ Then $\left(f_{0}, \xi_{0}\right)$ and $\left(f_{1}, \xi_{1}\right)$ are isotopic by an isotopy of tubular neighborhoods $F_{t} : E \rightarrow V, 0 \leqslant t \leqslant 1,$ such that $F_{0}=f_{0}$ and $$ F_{1}\left(D_{\varepsilon}\left(\xi_{0}\right)\right)=D_{\delta}\left(\xi_{1}\right), $$ where $D_{\varepsilon}\left(\xi_{0}\right)$ is the disk subbundle of $\xi_0$ of radius $\varepsilon$.