Local integral chains detect global natural maps into Chern targets

Apply normalized free integral chains and then sheafification. Its cohomology sheaves are computed from the local homology groups, so the hypothesis makes the resulting chain map a quasi-isomorphism of complexes of sheaves. This is an equivalence in the derived category, not an assertion that the original spaces are locally weakly equivalent.

The derived adjunction in [F2] identifies $\mathrm{Map}(Q,K(D,n))\to \mathrm{Map}(P,K(D,n))$ with the map of derived Hom spaces obtained by precomposing with that chain equivalence. It is an equivalence, hence injective on ${\pi }_0$. The adjunction and chain map are natural in $P,Q,D$, so this proves an identity in the local sheaf homotopy category. Evaluating a homotopy between fibrant natural maps gives the corresponding homotopy on derived sections over $X$; it retains all gluing data.

Use step 2.1 for $j$ and $j\times j$ by [F1]. A Cartan difference is a map to one such abelian Eilenberg target, so its vanishing on the free-class test forces its vanishing as the global natural map. No generation of $K_0(X)$ by global vector bundles is used.