@BalarkaSen Also, I have an argument you're gonna detest. I figured out how to get the Hurewicz theorem proper from the Serre SS. I'm sure you know that we get an abstract nonsense isomorphism $\pi_n(X)\cong H_n(X)$ for $(n-1)$-connected $X$ from the Serre SS by dimension shifting along the loop space fibration $\Omega X\rightarrow PX\rightarrow X$.
Now, the loop space fibration is natural in $X$ and both the Serre SS and the fibration LES are natural with respect to fibrations, so this abstract nonsense argument actually constructs a natural isomorphism $\pi_n(-)\cong H_n(-)$ of $\mathbf{A…