This is a reference request.
A theorem of Hurewicz (published in Beiträge zur Topologie der Deformationen. IV. Asphärische Räume, Proc. Akad. Wetensch. Amsterdam, volume 39, deel 2 (1936), 215-224, digitised version) asserts $\left[X,Y\right]=Hom(\pi_1X,\pi_1Y)/Inn(\pi_1Y)$ for aspherical spaces.
The generalization to arbitrary (pairs of) spaces reads: when $f,g:(X,X_1)\rightarrow (Y,Y_1)$ are such that $f_\ast=g_\ast:\pi_i(X)\rightarrow \pi_i(Y)$ and $f_\ast=g_\ast:\pi_i(X_1)\rightarrow \pi_i(Y_1)$ for all $i$, then $f$ and $g$ are homotopic (as maps of pairs).
I think I know how to prove this for pairs of CW complexes. My question is just for a citable reference either in a paper or a textbook.