Observing that every Tychonoff space X has an extension $kX$ which is a weakly Lindel$ddot{o}$f space and the minimal quasi-F cover $QF(kX)$ of $kX$ is a weakly Lindel$ddot{o}$f, we show that ${Phi}_{kX}:QF(kX){ ightarrow}kX$ is a $z^{sharp}$-irreducible map and that $QF({eta}X)=QF(kX)$. Using these, we prove that $QF(kX)=kQF(X)$ if and only if ${Phi}^k_X:kQF(X){ ightarrow}kX$ is an onto map and ${eta}QF(X)=(QF{eta}X)$.