Regarding the proof of Poincare Duality via 5-Lemma (commutativity?)

Question

The proof of Poincare Lemma for oriented manifolds with finite good cover (terminology of Bott, Tu) states that \begin{align} H^q (M) \simeq H^{n-q} _c (M) \end{align} where $n=\dim M$. The proof involves defining a pairing $\langle \cdot,\cdot \rangle: H^q (M) \otimes H^{n-q} _c (M) \to \mathbb{R}$ and shows that this pairing is non-degenerate (i.e. there is a iso $H^q (M) \simeq H^{n-q} _c (M)$) if $M$ is diffeomorphic to $\mathbb{R}^n$. Then one writes the Mayer-Vietoris sequence for the non-compact case and the dual Mayer-Vietoris for the compact case: $\require{AMScd}$ \begin{CD} H^{q-1}(U)\oplus H^{q-1}(V) @>>> H^{q-1}(U \cap V) @>>> H^q(U \cup V) @>>> H^q(U)\oplus H^q(V) @>>> H^q(U \cap V) \\ @VVV @VVV @VVV @VVV @VVV\\ H_c ^{n-(q-1)}(U) ^\ast \oplus H_c ^{n-(q-1)}(V)^\ast @>>> H_c ^{n-(q-1)}(U \cap V)^\ast @>>> H_c ^{n-q}(U \cup V)^\ast @>>> H_c ^{n-q}(U) ^\ast \oplus H_c ^{n-q}(V)^\ast @>>> H_c ^{n-q}(U \cap V)^\ast \end{CD} Then one calls the 5-Lemma for the case that $M=U \cup V$ and $\{U,V\}$ is a good cover. One proceeds by induction over the cardinality of the good cover of $M$. Am I right so far? My problem now is: The 5-Lemma applies if the diagramm commutes. But it does not; in fact the second box is only commutative up to a sign. (see Bott, Tu p.45) How come this is not a problem and how come nobody asked this before? Did I overlook something obvious? Thanks!