I'm reading the books by Bott & Tu and Milnor & Stasheef simultaneously. The following is my doubt:

The Thom isomorphism in Bott & Tu is obtained as $H_{cv}^{*+n}(E)\rightarrow H^*(M)$, where $\pi\colon E\to M$ is $n$ plane bundle over the manifold of dimension $m$ manifold $M$ and the isomorphism is given by the integration along fiber map $\pi_*$. By the projection formula, the Thom isomorphism $\mathscr{T}$, inverse to $\pi_*$, is then explicitly given by $\omega\mapsto \pi^*(\omega)\wedge\Phi$, where $\Phi=\mathscr{T}(1)$ is the Thom class of $E$.

In Milnor & Stasheef, the isomorphism is $\mathscr{T}\colon H^*(M)\to H^{*+n}(E,E_0)$, where $E_0$ is the complement of zero section and the map factors through $H^*(E)$ i.e. $H^*(M)\xrightarrow{\pi^*} H^*(E)\xrightarrow{\smile\text{fundamental class}} H^{*+n}(E,E_0)$.

I know that $H_{c}^{*+n}(E)=\varinjlim H_{c}^{*+n}(E, E-K)$ over the directed set of compact subsets of $E$. My question is, how can I relate $H^{*+n}(E,E_0)$ and $H_{cv}^{*+n}(E)$.

Any hint would be helpful. Thanks.