Proposition 4.3.19 (Spectrification). Let \(C\) be an \(\infty \)-category with finite limits and sequential colimits. Assume that \(\Omega \colon C_* \to C_*\) preserves sequential colimits. Then the inclusion \(\Sp (C) \hookrightarrow \PSp (C)\) admits a left adjoint \[ (-)^{\sp }\colon \PSp (C) \to \Sp (C), \qquad X \mapsto X^{\sp }, \] called the spectrification functor.
Proof. By Lemma 4.3.8, the limit of a diagram of spectra can first be computed in \(\PSp (C)\). Since \(\Omega \) preserves limits, the structure maps of this pointwise limit are again equivalences. Thus the limit prespectrum is a spectrum. □
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