Lemma 4.3.7. Let \(X\) and \(Y\) be prespectra in \(C\). Then the hom anima \(\Hom _{\PSp (C)}(X,Y)\) is the equalizer of the two maps \[ \Hom _{C_*^{\N }}(X,Y) \rightrightarrows \Hom _{C_*^{\N }}(X,\Omega \sh Y). \] The first map sends \(f\colon X\to Y\) to the composite \[ X \xrightarrow {f} Y \xrightarrow {\sigma ^Y} \Omega \sh Y, \] and the second sends it to the composite \[ X \xrightarrow {\sigma ^X} \Omega \sh X \xrightarrow {\Omega \sh (f)} \Omega \sh Y. \]
Proof. The projection \(\PSp (C) \to C_*^{\N }\) is conservative because a morphism in the arrow category is an isomorphism precisely when its source and target maps are isomorphisms. The projections \(C_*^{\N } \to C_*\) are jointly conservative by definition of the product. □
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