As an exercise, I'm trying to show that for an $(n-1)$-connected space $L$ with $\pi=\pi_n(L)$, the map $\iota_L:L\rightarrow K(\pi,n)$ associated to the fundamental class $\iota_L\in H^n(L;\pi)$ induces an isomorphism $\pi_n(L)\rightarrow \pi_n(K(\pi,n))$. After playing with this for a while, I've found that this feels so tautological that it *has* to be true, although that doesn't quite constitute a proof. I'm supposed to use only basic definitions / first principles. Presumably, these are:

The universal coefficient theorem yields $H^n(L;\pi)\cong Hom(H_n(L),\pi)$. The Hurewicz homomorphism $h:\pi \rightarrow H_n(L)$ is an isomorphism, and its inverse $h^{-1}\in Hom(H_n(L),\pi)$ corresponds to $\iota_L\in H^n(L;\pi)$.

We have a canonical bijection $H^n(L;\pi) \cong [L,K(\pi,n)]$. Any $[f]\in [L,K(\pi,n)]$ corresponds to $f^*\iota\in H^n(L;\pi)$, where $\iota\in H^n(K(\pi,n);\pi)$ is the fundamental class of $K(\pi,n)$ (which is associated to its identity map). This is actually a group isomorphism if we add maps on the right side by using the fact that $K(\pi,n)=\Omega K(\pi,n+1)$ (or at least $\simeq$, although what does $K(\pi,n)$ even mean really). I'd imagine that this is canonical too, but I don't know for sure.

We have a map $[L,K(\pi,n)]\rightarrow Hom(\pi,\pi)$ given by $[f]\mapsto f_\#$. Presumably the idea is to show that the image of $\iota_L$ is an isomorphism, but I can't tell if I'm just complicating the question by phrasing it in these terms.

I think my problem is that I don't really understand the defining way Eilenberg-MacLane spaces work. I have an intuitive picture of the Hurewicz homomorphism, and so I guess I have an intuitive picture of its inverse: it takes a homology class in degree $n$ and realizes it as the image of a bunch of based $n$-spheres (which is made possible by the Hurewicz theorem). We can look at this more or less as a cochain in $C^n_{cell}(L;\pi)$, and this is the element $\iota_L\in H^n(L;\pi)$. But then I have no idea how to actually turn this into a map $\iota_L : L\rightarrow K(\pi,n)$, or whether I'm even supposed to.

Because I'd still like to work this out myself to whatever extent I can, a good hint (if one exists) is worth more to me than a straight-up answer. Of course, I'm happy with either. Thanks!