In Corollary 4.24 (Freudenthal Susepension Theroem) of Hatcher - Algebraic Topology, Hatcher says that the suspension map $S:\pi_i(X)\to \pi_{i+1}(SX)$ is same as the map $\pi_i(X)\cong \pi_{i+1}(C_+X,X)\to \pi_{i+1}(SX,C_-X)\cong \pi_{i+1}(SX)$. (Here $C_+X$ and $C_-X$ are respectively the upper and lower cones in $SX$, the isomorphisms are from homotopy long exact sequences (since cones are contractible), and the middle map is induced by inclusion.) Is the composition the definition of $S$? Or is the following correct? Every basepoint-preserving map $f:S^i\to X$ induces a basepoint-preserving map $Sf:SS^{i}=S^{i+1}\to SX$ and $S$ is defined by $[f]\mapsto [Sf]$.
I think the latter seems correct, but I can't see why the latter map is the same as the composition map $\pi_i(X)\cong \pi_{i+1}(C_+X,X)\to \pi_{i+1}(SX,C_-X)\cong \pi_{i+1}(SX)$. Am I missing something?