I am trying to understanding the construction of the Spanier-Whitehead category, and the role of the Freudenthal Suspension Theorem in this.
FST seems to take many forms, but here is the one that I think is being used to construct the SW category:
Let $X$ and $Y$ be pointed CW complexes, $n\geq 0$. Suppose that $X$ is $n$-connected and $\text{dim}(Y)\leq 2n$. Then the map $[X,Y]\to[\Sigma X,\Sigma Y]$ induced of the reduced suspension functor $\Sigma$ is a bijection.
Let $X$ and $Y$ be finite pointed CW-complexes. Then we have a diagram $$[\Sigma^2 X,\Sigma^2 Y]\to[\Sigma^3 X,\Sigma^3 Y]\to\cdots$$ in the category of abelian groups, with colimit $[\Sigma^\infty X,\Sigma^\infty Y]$.
I am trying to understand specifically how the Freudenthal Suspension Theorem implies that the group homomorphisms $$[\Sigma^N X,\Sigma ^N Y]\to[\Sigma^\infty X,\Sigma^\infty Y]$$ are isomorphisms for sufficiently large $N$.
For each pointed space $X$, there is a canonical basepoint-preserving map $X\to\Omega(\Sigma X)$. For pointed spaces $X$ and $Y$, there is a canonical group isomorphism $$[\Sigma X,Y]\to[X,\Omega Y]$$ (although I'm not quite sure about the group structure on $[X,\Omega Y]$). Applying this to $\Sigma^k X$ and $\Sigma^{k+1} Y$, we get an isomorphism $$[\Sigma^{k+1}X,\Sigma^{k+1}Y]\to[\Sigma^k X,\Omega(\Sigma^{k+1}Y)]$$ for $k\geq 0$... then somehow we obtain isomorphisms $$[\Sigma^k X,\Sigma^k Y]\to[\Sigma^k X,\Omega(\Sigma^{k+1}Y)]$$ via Freudenthal's Suspension Theorem, hence isomorphisms $[\Sigma^k X,\Sigma^k Y]\to[\Sigma^{k+1}X,\Sigma^{k+1}Y]$. I'm sure I'm missing something or misunderstood it. How are isomorphisms $[\Sigma^n X,\Sigma^n Y]\to[\Sigma^{n+1}X,\Sigma^{n+1}Y]$ obtained for sufficiently large $n$?
It looks like the main source for this version of FST is Corollary 3.2.3 in Bordism, stable homotopy and Adams spectral sequences by Stanley Kochman, but I can't get hold of this. It is cited in these notes without proof.
For any $X$, $\Sigma^k X$ is $(k-1)$-connected for any $k$. Also, if $\dim Y = d$, then $\dim \Sigma^k Y = d + k$, and $d + k \leq 2(k-1)$ if $d \leq k-2$. So if $d \leq k-2$, then the Freudenthal Suspension Theorem applies to $[\Sigma^k X, \Sigma^k Y] \to [\Sigma^{k+1} X, \Sigma^{k+1} Y]$, and indeed to all of the maps after that: $$ [\Sigma^k X, \Sigma^k Y] \to [\Sigma^{k+1} X, \Sigma^{k+1} Y] \to [\Sigma^{k+2} X, \Sigma^{k+2} Y] \to \dots $$ are all bijections, and in fact all group isomorphisms (assuming that $k > 0$ so that these sets all have group structures). So they are all isomorphic to $[\Sigma^\infty X, \Sigma^\infty Y]$.