Let $X$ and $Y$ be CW complexes (resp. Kan complexes) and let $f : X \to Y$ be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result:

Theorem.The following are equivalent:

- $f : X \to Y$ is a homotopy equivalence.
- $f_* : \pi_0 (X) \to \pi_0 (Y)$ is a bijection; and for all $n \ge 1$ and all points $x$, $f_* : \pi_n (X, x) \to \pi_n (Y, f (y))$ is an isomorphism.
- $f_* : \pi_{\le 1} (X) \to \pi_{\le 1} (Y)$ is an equivalence of groupoids; and for all $n \ge 0$ and all locally constant abelian sheaves (or equivalently, local systems of abelian groups) $\mathscr{A}$ on $Y$, $f^* : H^n (Y, \mathscr{A}) \to H^n (X, f^* \mathscr{A})$ is an isomorphism.

The equivalence of conditions (1) and (2) is the usual Whitehead theorem; I am interested in the equivalence of (2) and (3). Unfortunately, I have not been able to find a proof in the literature. Can someone provide a reference or a sketch proof?