For connected CW complexes, weak homotopy equivalence already means homotopy equivalence.Suggest a slogan
Let \(f\colon X\to Y\) be a map of connected CW complexes that induces isomorphisms \(f_*\colon\pi_n(X)\to\pi_n(Y)\) for all \(n\), that is, a weak homotopy equivalence. Then \(f\) is a homotopy equivalence [@Hat02].