https://www.maths.ed.ac.uk/~v1ranick/papers/botttu.pdf
This is a question about the following proposition in the book.
Proposition 10.6. Let $\Bbb R$ be the constant presheaf on a manifold $M$. Then the Čech cohomology of $M$ with values in $\Bbb R$ is isomorphic to the de Rham cohomology.
Proof) Since the good covers are cofinal in the set of all covers of $M$ (Corollary 5.2), we can use only good covers in the direct limit $H^*(M,\Bbb R)=\lim_{\mathfrak{U}}H^*(\mathfrak{U},\Bbb R)$. By Theorem 8.9, $H^*(\mathfrak{U},\Bbb R) \cong H^*_{DR}(M)$ for any good cover of M. Moreover, it is easily seen that this isomorphism is compatible with refinement of good covers. Therefore, there is an isomorphism $H^*(M,\Bbb R) \cong H^*_{DR}(M)$.
How it is easily seen that the isomorphism $H^*(\mathfrak{U},\Bbb R) \cong H^*_{DR}(M)$ is compatible with refinement of good covers?
If $\mathfrak{U}, \mathfrak{V}$ are two good covers of $M$ with $\mathfrak{V}$ a refinement of $\mathfrak{U}$, then there is a well-defined map $H^*(\mathfrak{U},\Bbb R)\to H^*(\mathfrak{V},\Bbb R)$ defined above Lemma 10.4.1. Also the formula of an explicit isomorphism $H^*(\mathfrak{U},\Bbb R) \cong H^*_{DR}(M)$ is given in Proposition 9.8. But I can't see why these maps should be compatible.
If you think of the double complex $C^*(\mathfrak U, \Omega^*)$ as forming a rectangular array (as the book does) and set $C^*(\mathfrak V,\Omega^*)$ right underneath it, you get commutative squares vertically in both directions (i.e., $d$ and $\delta$ both commute with restriction). This implies that the collating formula restricts correctly, and I believe that's all you need.