Cohomology represented by a spectrum

Definition 3.2.3 (\(E\)-cohomology). Let \(E\) be a spectrum and let \(X \in \An _*\) be a pointed anima. For \(k \in \Z \), we define the reduced \(E\)-cohomology of \(X\) by \[ \widetilde E^k(X) := [X,E_k]_*. \] The isomorphism \(E_k\iso \Omega ^2E_{k+2}\) identifies this set with the second homotopy group of the pointed mapping anima \(\Hom _{\An _*}(X,E_{k+2})\), and therefore equips it with a natural abelian group structure. The structure isomorphisms of \(E\) induce natural isomorphisms \[ \widetilde E^k(X)=[X,E_k]_* \xrightarrow {\cong } [X,\Omega E_{k+1}]_* \xrightarrow {\cong } [\Sigma X,E_{k+1}]_*=\widetilde E^{k+1}(\Sigma X), \] which define the suspension isomorphism for \(\widetilde E^*\). Both maps are group homomorphisms: the first is induced by an isomorphism of pointed animae, while the second comes from the natural equivalence \(\Hom _{\An _*}(\Sigma X,Y)\simeq \Hom _{\An _*}(X,\Omega Y)\) of Lemma 2.4.8, which induces isomorphisms on the homotopy groups of mapping animae.