Let $b(d)$ be multiplicative function defined on squarefree numbers such that
$$ r_\mathcal A(d)=|\mathcal A_d|-{b(d)\over d}X $$
relatively small, and $P(z)$ is the product of primes in $\mathcal P\cap[2,z)$. If there exists $\kappa,A>0$ such that whenever $2\le w<z$
$$ \prod_{\substack{p\in\mathcal P\\w\le p<z}}\left(1-{b(p)\over p}\right)^{-1}\le\left(\log z\over\log w\right)^\kappa\left(1+{A\over\log w}\right) $$
then the fundamental lemma of the combinatorial sieve states that
$$ S(\mathcal A,\mathcal P,z)=XV(z)\left\{1+\mathcal O(u^{-u/2})\right\}+\theta\sum_{\substack{d|P(z)\\d<z^u}}|r_\mathcal A(d)|\tag1 $$
where $|\theta|\le1$,
$$ V(z)=\prod_{\substack{p\in\mathcal P\\p<z}}\left(1-{b(p)\over p}\right) $$
and the $\mathcal O$-constant may depend on $\kappa$ and $A$.
The first time I encounter this is from Tenenbaum's Introduction to Analytic Number Theory, which states that an expository account of this technology is available in Halberstam & Richert's Sieve Methods. Nevertheless, after reading through this literature, I did not find any types of fundamental lemma similar to (1). Thus, I look up Diamond & Halberstam's A Higher-Dimensional Sieve Method but only found a version of fundamental lemma with the remainder term of Selberg's sieve (see Theorem 4.1 of that book):
$$ S(\mathcal A,\mathcal P,z)=XV(z)\left\{1+\mathcal O(e^{-u\log u-3u/2})\right\}+\theta\sum_{\substack{d|P(z)\\d<z^{2u}}}3^{\omega(d)}|r_\mathcal A(d)|\tag2 $$
Thus, I would love to know where I can find a full derivation of the fundamental lemma as presented in (1).