I am learning Postnikov towers from this lecture. Here is the first part of the proof that I am studying

Why is true the marked statement?

For example, let be $X = S^2$.

To build $Y_1$ (i.e, with killed $\pi_2$, $, \pi_3$...) I add to $X$ a $3$-cell glued on a generator of $\pi_2(S^2) = \mathbb{Z}\cdot[id]$. So $Y_1^{(1)} = D^3$. We have no more homotopy groups to kill, then $Y_1 = Y_1^{(1)} = D^3$.

To build $Y_2$ I add to $X$ a $4$-cell to a generator of $\pi_3(S^2) = \mathbb{Z}\cdot[hopf]$ and then maybe $5$-cells, $6$-cells...

Then what is the "canonical inclusion" $Y_2 \to Y_1$? I had not adjoined more cells for $Y_1$ than for $Y_2$ as claimed!

continuous map from $S^{k}$ to $Y_n^{(k-1)}$. This is the canonical thing to do (it is functorial as it involves no choices). With this construction it is indeed true that $Y_{n+1}$ is contained in $Y_n$ for the reason given. $\endgroup$ – Gustavo Granja Jan 18 at 17:40each