Unreduced suspension spectra

Proof. First define the free prespectrum functor \[ \Sigma ^{\infty ,\pre }\colon C_* \to \PSp (C) \] by \[ \Sigma ^{\infty ,\pre }(X)_n := \Sigma ^n X, \] with structure maps \(\Sigma ^nX \to \Omega \Sigma ^{n+1}X\) given by the unit of the adjunction \(\Sigma \dashv \Omega \) in \(C_*\). We claim that \(\Sigma ^{\infty ,\pre }\) is left adjoint to evaluation at level zero \(\ev _0\colon \PSp (C) \to C_*\). Indeed, a map \(\Sigma ^{\infty ,\pre }(X) \to Y\) is determined by its level-zero component \(X \to Y_0\): once \(f_n\colon \Sigma ^nX \to Y_n\) is known, compatibility with the structure maps forces \(f_{n+1}\colon \Sigma ^{n+1}X \to Y_{n+1}\) to be adjoint to the composite \[ \Sigma ^nX \xrightarrow {f_n} Y_n \xrightarrow {\sigma ^Y_n} \Omega Y_{n+1}. \] Conversely, this recursive construction produces a compatible map of prespectra. The construction is natural in \(X\) and \(Y\) and applies equally to parametrized families of maps, so it gives an isomorphism of hom animae \[ \Hom _{\PSp (C)}(\Sigma ^{\infty ,\pre }X,Y) \simeq \Hom _{C_*}(X,Y_0). \]

Now set \[ \Sigma ^{\infty }(X) := (\Sigma ^{\infty ,\pre }X)^{\mathrm {sp}}. \] For \(Y \in \Sp (C)\), spectrification gives natural equivalences \[ \Hom _{\Sp (C)}(\Sigma ^{\infty }X,Y) \simeq \Hom _{\PSp (C)}(\Sigma ^{\infty ,\pre }X,Y) \simeq \Hom _{C_*}(X,\Omega ^{\infty }Y), \] which proves the adjunction. □