Reduced and unreduced suspension spectra

Remark 4.4.3. Given a pointed anima \((X,x) \in \An _*\), note that we may write \(X\) as the cofiber of the basepoint inclusion \(*_+ \to X_+\). Since \(\Sigma ^{\infty }(-)\) preserves cofibers, we may write the reduced suspension spectrum of \(X\) as \[ \Sigma ^{\infty }(X,x) \quad \simeq \quad \cofib (\S \xrightarrow {x} \S [X]). \]