I am reading Lurie's notes about chromatic homotopy theory. In Lecture 22, he considers the p-local complex bordism spectrum $\operatorname{MU}^p$ with $\pi_*(\operatorname{MU}^p)=\mathbb{Z}_p [t_1,t_2,...]$. He defines $M(k)$ to be the cofiber of the map $\Sigma^{2k}\operatorname{MU}^p \to \operatorname{MU}^p$ given by multiplication by $t_k$ for every nonnegative k. He claims that $M(k)$ has a homotopy associative multiplication. 
Now I can not understand this proof.
First, why does he need $K$ to give this multiplication. I think that it can be given directly from the universal property of total homotopy fiber in the first square. You just consider the square with the down right corner element as $M(k)$ and other three elements as point. Then we can have a natural map from the first square in the proof to the sqaure to the square I build. By universal property of total homotopy fiber, we have the multiplication map.
Also, I want to know why he can check if the composition is nullhomotopic by homotopy group. It seems that you can not do this for topological space.