Let $p \colon E \to M$ be a smooth real vector bundle of rank $n$, $R$-oriented for a commutative ring $R$.
Recall that the Thom isomorphism theorem states that there is a unique class $u \in H^n(E, E \setminus M)$ such that for all $x \in M$, the restriction $$H^n(E, E \setminus M;R) \to H^n(E_x, E_x \setminus \{0\};R)$$ maps $u$ to the prefered generator given by the orientation. And furthermore $$ H^k(M;R) \to H^{k+n}(E, E \setminus M;R), \qquad \alpha \mapsto p^*(\alpha) \cup u $$ is an isomorphism.
Question: According to exercise 6.20 in Differential Forms in Algebraic Topology by Bott & Tu (p.65), there is also an isomorphism $$ H_c^k(M;R) \xrightarrow{\cong} H_c^{k+n}(E;R) $$ in cohomology with compact support given by the same formula. Can this be concluded from the above usual Thom theorem?