Dirichlet convolutions and a formula given in Selberg's sieve

Question

I was reading the Selberg's sieve theorem and stumbled on one equation that I honestly cannot understand. As in theorem, we let $A$ be a set of positive integers, $\mathcal{P}$ -set of primes, let $$ P(z) = \prod_{\substack{p \in \mathcal{P}\\ p < z}}p, $$ define $$ A_d := \{a \colon a \in A, d | a \} $$ and define completely multiplicative function $f \colon \mathbb{N} \to (0,1]$ to estimate $|A_{d}|$, i.e. $$ |A_{d}| = |A|f(d) + r(d). $$ Moreover, define functions $g, F$ and $V$ to be $$ g(n) = (\mu * f)(n) = \sum_{d|n}\mu(d)f\left(\frac{n}{d}\right) $$ $$ F(n) = \left(\frac{1}{f} * \mu\right) (n) = \sum_{d|n} \frac{\mu(d)}{f\left(\frac{n}{d}\right)} $$ $$ V(z) = \sum_{\substack{d \leq z \\ d|P(z)}} \frac{\mu^2(d)}{g(d)}. $$ One place in the proof of Selberg's formula requires the truth of formula $$ \sum_{\substack{d|P(z) \\ d \leq z}}\mu^2(d)F(d) = V(z). $$ I believe it is equivalent to proving that $$ \mu^2(n) \left( F(n) - \frac{1}{g(n)} \right) = 0. $$ But why is that?