Cohomological Whitehead theorem 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?
 A: The simplicial version of the result appears as part of Proposition 4 in [Quillen, Homotopical algebra, Ch. II §3]. Curiously, the proof is essentially the same as the one sketched by Justin Young above.
A: One of the uses of the local coefficient or covering space version of the arguments sketched above are in conjunction with Acyclic Model arguments, and there have been occasions when authors have initially neglected to move to covering spaces to obtain a homotopy rather than a homology  equivalence.  However Section 10.3.1 of the book Nonabelian Algebraic Topology (EMS Tracts in Mathematics vol 15, 2011) has a version of Acyclic Models based on crossed complexes, rather than chain complexes, and this yields homotopy equivalances, by a direct use of Whitehead's theorem. 
Also Section 12.4 is on ``Local Coefficients and Local Systems". Relations between crossed complexes and chain complexes with a groupoid of operators are spelled out over several subsections of 7.4 and 9.5. See also Section 8.4. 
