Do these conditions imply homotopy equivalence?

Question

Suppose we have maps $f:Z\leftrightarrows X:g$ both of which are $\pi_*$ isomorphisms and satisfy $f\circ g \simeq \operatorname{id}_X$. Suppose also that $Z$ is a CW complex.

Question: Do the maps form a homotopy equivalence?

Answer 1

Yes, they do.

The composition $f\circ g$ is homotopic to identity map of $X$ (because it already identity).

Note that $\pi_*(f)=\pi_*(g)^{-1}$, therefore $\pi_*(g\circ f)=\mathrm{Id}_{\pi_*(Z)}$. So $g\circ f$ is homotopic to $\mathrm{Id}_Z$ (using the fact that $Z$ is $CW$-complex).